Crystallization

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Crystallization

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Crystallization Fourth Edition

J. W. Mullin Emeritus Professor of Chemical Engineering, University of London

OXFORD BOSTON JOHANNESBURG MELBOURNE NEW DELHI SINGAPORE

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Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd A member of the Reed Elsevier plc group First published 1961 Second edition 1972 Third edition 1992 Reprinted 1994, 1995 Paperback edition 1997 Fourth edition 2001 # Reed Educational and Professional Publishing Ltd 2001

All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers

British Library Cataloguing in Publication Data A Catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A Catalogue record for this book is available from the Library of Congress ISBN 0 7506 4833 3

Typeset in India at Integra Software Services Pvt Ltd, Pondicherry, India 605005; www.integra-india.com

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Contents Preface to Fourth Edition Preface to First Edition Nomenclature and units

viii x xii

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

The crystalline state Liquid crystals Crystalline solids Crystal symmetry Crystal systems Miller indices Space lattices Solid state bonding Isomorphs and polymorphs Enantiomorphs and chirality Crystal habit Dendrites Composite crystals and twins Imperfections in crystals

1 1 3 4 7 10 13 15 16 18 22 24 25 27

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

Physical and thermal properties Density Viscosity Surface tension Diffusivity Refractive index Electrolytic conductivity Crystal hardness Units of heat Heat capacity Thermal conductivity Boiling, freezing and melting points Enthalpies of phase change Heats of solution and crystallization Size classification of crystals

32 32 35 39 41 47 48 48 49 50 54 55 58 62 64

3 3.1 3.2 3.3 3.4 3.5

Solutions and solubility Solutions and melts Solvent selection Expression of solution composition Solubility correlations Theoretical crystal yield

86 86 86 90 92 96

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vi

Contents

3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Ideal and non-ideal solutions Particle size and solubility Effect of impurities on solubility Measurement of solubility Prediction of solubility Solubility data sources Supersolubility Solution structure

98 108 110 112 120 123 123 132

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Phase equilibria The phase rule One-component systems Two-component systems Enthalpy±composition diagrams Phase change detection Three-component systems Four-component systems `Dynamic' phase diagrams

135 135 136 139 146 151 156 169 179

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Nucleation Primary nucleation Secondary nucleation Metastable zone widths Effect of impurities Induction and latent periods Interfacial tension Ostwald's rule of stages

181 182 195 201 205 206 210 214

6 6.1 6.2 6.3 6.4 6.5 6.6

Crystal growth Crystal growth theories Growth rate measurements Crystal growth and dissolution Crystal habit modification Polymorphs and phase transformations Inclusions

216 216 236 260 269 280 284

7 7.1 7.2 7.3 7.4 7.5 7.6

Recrystallization Recrystallization schemes Resolution of racemates Isolation of polymorphs Recrystallization from supercritical fluids Zone refining Single crystals

289 289 295 300 302 303 309

8 Industrial techniques and equipment 8.1 Precipitation 8.2 Crystallization from melts

315 315 343

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Contents

vii

8.3 Sublimation 8.4 Crystallization from solution

358 368

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7

403 403 430 434 451 459 463 467

Crystallizer design and operation Crystal size distribution (CSD) Kinetic data measurement and utilization Crystallizer specification Fluid±particle suspensions Encrustation Caking of crystals Downstream processes

Appendix References Author index Subject index

478 536 577 587

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Preface to Fourth Edition This fourth edition of Crystallization has been substantially rewritten and up-dated. The 1961 first edition, written primarily for chemical engineers and industrial chemists, was illustrated with practical examples from a range of process industries, coupled with basic introductions to the scientific principles on which the unit operation of crystallization depends. It was also intended to be useful to students of chemical engineering and chemical technology. The aims and objectives of the book have remained intact in all subsequent editions, although the subject matter has been considerably expanded each time to take into account technological developments and to reflect current research trends into the fundamentals of crystallization mechanisms. The continuing upsurge in interest in the utilization of crystallization as a processing technique covers an increasing variety of industrial applications, not only in the long-established fields of bulk inorganic and organic chemical production, but also in the rapidly expanding areas of fine and specialty chemicals and pharmaceuticals. These developments have created an enormous publication explosion over the past few decades, in a very wide range of journals, and justify the large number of specialist symposia that continue to be held world-wide on the subject of crystallization. Particular attention is drawn in this edition to such topical subjects as the isolation of polymorphs and resolution of enantiomeric systems, the potential for crystallizing from supercritical fluids, the use of molecular modelling in the search for tailored habit modifiers and the mechanisms of the effect of added impurities on the crystal growth process, the use of computer-aided fluid dynamic modelling as a means of achieving a better understanding of mixing processes, the separate and distinct roles of both batch and continuous crystallization processing, and the importance of potential downstream processing problems and methods for their identification from laboratory investigations. Great care has been taken in selecting suitable literature references for the individual sections to give a reliable guide to further reading. Once again I want to record my indebtedness to past research students, visiting researchers and colleagues in the Crystallization Group at University College London over many years, for their help and support in so many ways. They are too numerous to name individually here, but much of their work is recorded and duly acknowledged in appropriate sections throughout this edition. I should like to express my sincere personal thanks to them all. I am also very grateful to all those who have spoken or written to me over the years with useful suggestions for corrections or improvements to the text. Finally, and most importantly, it gives me great pleasure to acknowledge the debt I owe to my wife, Averil, who has assisted me with all four editions of

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Preface to Fourth Edition

ix

Crystallization. Without her tremendous help in preparing the manuscripts, my task of writing would not have been completed. JOHN MULLIN University College London 2001

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Preface to First Edition Crystallization must surely rank as the oldest unit operation, in the chemical engineering sense. Sodium chloride, for example, has been manufactured by this process since the dawn of civilization. Today there are few sections of the chemical industry that do not, at some stage, utilize crystallization as a method of production, purification or recovery of solid material. Apart from being one of the best and cheapest methods available for the production of pure solids from impure solutions, crystallization has the additional advantage of giving an end product that has many desirable properties. Uniform crystals have good flow, handling and packaging characteristics: they also have an attractive appearance, and this latter property alone can be a very important sales factor. The industrial applications of crystallization are not necessarily confined to the production of pure solid substances. In recent years large-scale purification techniques have been developed for substances that are normally liquid at room temperature. The petroleum industry, for example, in which distillation has long held pride of place as the major processing operation, is turning its attention most keenly to low-temperature crystallization as a method for the separation of `difficult' liquid hydrocarbon mixtures. It is rather surprising that few books, indeed none in the English language, have been devoted to a general treatment of crystallization practice, in view of its importance and extensive industrial application. One reason for this lack of attention could easily be that crystallization is still referred to as more of an art than a science. There is undoubtedly some truth in this old adage, as anyone who has designed and subsequently operated a crystallizer will know, but it cannot be denied that nowadays there is a considerable amount of science associated with the art. Despite the large number of advances that have been made in recent years in crystallization technology, there is still plenty of evidence of the reluctance to talk about crystallization as a process divorced from considerations of the actual substance being crystallized. To some extent this state of affairs is similar to that which existed in the field of distillation some decades ago when little attempt had been made to correlate the highly specialized techniques developed, more or less independently, for the processing of such commodities as coal tar, alcohol and petroleum products. The transformation from an `art' to a `science' was eventually made when it came to be recognized that the key factor which unified distillation design methods lay in the equilibrium physical properties of the working systems. There is a growing trend today towards a unified approach to crystallization problems, but there is still some way to go before crystallization ceases to be the Cinderella of the unit operations. More data, particularly of the applied kind, should be published. In this age of prolific outputs of technical literature such a recommendation is not made lightly, but there is a real deficiency of this type

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xi

of published information. There is, at the same time, a wealth of knowledge and experience retained in the process industries, much of it empirical but none the less valuable when collected and correlated. The object of this book is to outline the more important aspects of crystallization theory and practice, together with some closely allied topics. The book is intended to serve process chemists and engineers, and it should prove of interest to students of chemical engineering and chemical technology. While many of the techniques and operations have been described with reference to specific processes or industries, an attempt has been made to treat the subject matter in as general a manner as possible in order to emphasize the unit operational nature of crystallization. Particular attention has been paid to the newer and more recently developed processing methods, even where these have not as yet proved adaptable to the large-scale manufacture of crystals. My thanks are due to the Editors of Chemical Engineering Practice for permission to include some of the material and many of the diagrams previously published by me in Volume 6 of their 12-volume series. I am indebted to Professor M. B. Donald, who first suggested that I should write on this subject, and to many of my colleagues, past and present, for helpful discussions in connection with this work. I would also like to take this opportunity of acknowledging my indebtedness to my wife for the valuable assistance and encouragement she gave me during the preparation of the manuscript. JOHN MULLIN London 1960

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Nomenclature and units The basic SI units of mass, length and time are the kilogram (kg), metre (m) and second (s). The basic unit of thermodynamic temperature is the kelvin (K), but temperatures and temperature differences may also be expressed in degrees Celsius ( C). The unit for the amount of substance is the mole (mol), defined as the amount of substance which contains as many elementary units as there are atoms in 0.012 kg of carbon-12. Chemical engineers, however, are tending to use the kilomole (kmol ˆ 103 mol) as the preferred unit. The unit of electric current is the ampere (A). Several of the derived SI units have special names: Quantity

Name

Frequency Force Pressure Energy, work; heat Power Quantity of electricity Electric potential Electric resistance Conductance Capacitance Magnetic flux Magnetic flux density Inductance

hertz newton pascal joule watt coulomb volt ohm siemens farad weber tesla henry

Symbol Hz N Pa J W C V

S F Wb T H

SI unit

Nm Nm Js 1

2

WA 1 VA 1 AV 1 CV 1 Vs Wb m 2 Wb A 1

Basic SI unit s 1 m kg s 2 m 1 kg s 2 m2 kg s 2 m2 kg s 3 sA m2 kg s 3 A 1 m2 kg s 3 A 2 m 2 kg 1 s3 A2 m 2 kg 1 s4 A2 m2 kg s 2 A 1 kg s 2 A 1 m2 kg s 2 A 2

Up to the present moment, there is no general acceptance of the pascal for expressing pressures in the chemical industry; many workers prefer to use multiples and submultiples of the bar (1 bar ˆ 105 Pa ˆ 105 N m 2  1 atmosphere). The standard atmosphere (760 mm Hg) is defined as 1:0133  105 Pa, i.e. 1.0133 bar. The prefixes for unit multiples and submultiples are: 10 10 10 10 10 10 10 10

18 15 12 9 6 3 2 1

atto femto pico nano micro milli centi deci

a f p n m m c d

101 102 103 106 109 1012 1015 1018

deca hecto kilo mega giga tera peta exa

da h k M G T P E

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Nomenclature and units

xiii

Conversion factors for some common units used in chemical engineering are listed below. An asterisk ( ) denotes an exact relationship. Length



1 in 1 ft  1 yd  1 mile  1 AÊ (aÊngstrom)

: : : : :

25.4 mm 0.3048 m 0.9144 m 1.6093 km 10 10 m

Time



1 min 1h  1 day  1 year

: : : :

60 s 3.6 ks 86.4 ks 31.5 Ms

Area



1 in2 1 ft2  1 yd2  1 acre  1 hectare  1 mile2

: : : : : :

645:16 mm2 0:092903 m2 0:83613 m2 4046:9 m2 10 000 m2 2:590 km2



1 in3 1 ft3  1 yd3  1 UK gal  1 US gal

: : : : :

16:387 cm3 0:02832 m3 0:76453 m3 4546:1 cm3 3785:4 cm3

Mass



1 oz 1 grain  1 lb  1 cwt  1 ton

: : : : :

28.352 g 0.06480 g 045359237 kg 508023 kg 1016.06 kg

Force



1 pdl 1 lbf  1 kgf  1 tonf  1 dyn

: : : : :

0.13826 N 4.4482 N 9.8067 N 9.9640 kN 10 5 N

Temperature difference



:

5 9

Energy (work, heat)



: : : : : : : : : :

1.3558 J 0.04214 J 4.1868 J 10 7 J 1.05506 kJ 1.8991 kJ 2.6845 MJ 3.6 MJ 105.51 MJ 4.1855 MJ

Volume













1 degF (degR)

1 ft lbf 1 ft pdl  1 cal (internat. table)  1 erg  1 Btu  1 chu  1 hp h  1 kW h  1 therm  1 thermie 

degC (K)

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xiv

Nomenclature and units

Calorific value (volumetric)



1 Btu/ft3 1 chu/ft3  1 kcal/ft3  1 kcal/m3  1 therm/ft3

: : : : :

37:259 kJ m 3 67:067 kJ m 3 147:86 kJ m 3 4:1868 kJ m 3 3:7260 GJ m 3





Velocity

1 ft/s 1 ft/min  1 ft/h  1 mile/h

: : : :

0:3048 m s 1 5:0800 mm s 1 84:667 mm s 1 0:44704 m s 1

Volumetric flow



1 ft3 /s 1 ft3 /h  1 UK gal/h  1 US gal/h

: : : :

0:028316 m3 s 1 7:8658 cm3 s 1 1:2628 cm3 s 1 1:0515 cm3 s 1



: :

0:12600 g s 1 0:28224 kg s 1

1 lb/in2 1 lb/ft2  1 ton/mile2

: : :

703:07 kg m 2 4:8824 kg m 2 392:30 kg km 2



1 lb/in3 1 lb/ft3  1 lb/UK gal  1 lb/US gal

: : : :

27:680 g cm 16:019 kg m 99:776 kg m 119:83 kg m



Mass flow Mass per unit area

Density







1 lb/h 1 ton/h







3 3 3 3

Pressure

1 lbf/in2 1 tonf/in2  1 lbf/ft2  1 kgf/m2  1 standard atm  1 at (1 kgf/cm2 )  1 bar  1 ft water  1 in water  1 inHg  1 mmHg (1 torr)

: : : : : : : : : : :

6:8948 kN m 2 15:444 MN m 2 47:880 N m 2 9:8067 N m 2 101:325 kN m 2 98:0665 kN m 2 105 N m 2 2:9891 kN m 2 249:09 N m 2 3:3864 kN m 2 133:32 N m 2

Power (heat flow)



: : : : : : : : : :

745.70 W 735.50 W 10 7 W 1.3558 W 0.29308 W 1.0551 kW 0.52754 W 1.8991 kW 1.1630 kW 3516.9 W



1 hp (British) 1 hp (metric)  1 erg/s  1 ft lbf/s  1 Btu/h  1 Btu/s  1 chu/h  1 chu/s  1 kcal/h  1 ton of refrigeration 

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xv

Moment of inertia



1 lb ft2

:

0:042140 kg m2

Momentum



1 lb ft/s

:

0:13826 kg m s

Angular momentum



1 lb ft2 /s

:

0:042140 kg m2 s

Viscosity, dynamic



1 poise (1 g/cm s)

:



1 lb/ft h 1 lb/ft s

: :

0:1 N s m 2 (0:1 kg m 1 s 1 ) 0:41338 mN s m 2 1:4882 N s m 2

1 stokes (1 cm2 /s) 1 ft2 /h

: :

10 4 m2 s 1 0:25806 cm2 s

1 erg/cm2 (1 dyn/cm)

:

10 3 J m 2 (10 3 N m 1 )



Viscosity, kinematic

  

Surface per unit volume



1 ft2 /ft3

:

3:2808 m2 m

Surface per unit mass



1 ft2 /lb

:

0:20482 m2 kg

Mass flux density



1 lb/h ft2

:

1:3562 g s



1 Btu/h ft2 1 kcal/h m2

: :

3:1546 W m 2 1:163 W m 2

1 Btu/h ft2 F 1 kcal/h m2 C

: :

5:6784 W m 1:1630 W m

Heat flux density



1

1

Surface energy (surface tension)

1

1

3 1 2

m

Heat transfer coefficient



Specific enthalpy (latent heat, etc.)



1 Btu/lb

:

2:326 kJ kg

Heat capacity (specific heat)



1 Btu/lb F

:

4:1868 kJ kg



1 Btu/h ft F 1 kcal/h m C

: :

1:7307 W m 1 K 1 1:163 W m 1 K 1

Thermal conductivity





2 2

K K

1 1

1

1

K

1

The values of some common physical constants in SI units include: Avogadro number, NA Boltzmann constant, k Planck constant, h Stefan±Boltzmann constant,  Standard temperature and pressure (s.t.p.) Volume of 1 kmol of ideal gas at s.t.p. Gravitational acceleration Universal gas constant, R Faraday constant, F

6:023  1023 mol 1 1:3805  10 23 J K 1 6:626  10 34 J s 5:6697  10 8 W m 2 K 4 273:15 K and 1:013  105 N m 22:41 m3 9:807 m s 2 8:3143 J mol 1 K 1 9:6487  104 C mol

1

2

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1

The crystalline state

The three general states of matter ± gaseous, liquid and solid ± represent very different degrees of atomic or molecular mobility. In the gaseous state, the molecules are in constant, vigorous and random motion; a mass of gas takes the shape of its container, is readily compressed and exhibits a low viscosity. In the liquid state, random molecular motion is much more restricted. The volume occupied by a liquid is limited; a liquid only takes the shape of the occupied part of its container, and its free surface is flat, except in those regions where it comes into contact with the container walls. A liquid exhibits a much higher viscosity than a gas and is less easily compressed. In the solid state, molecular motion is confined to an oscillation about a fixed position, and the rigid structure generally resists compression very strongly; in fact it will often fracture when subjected to a deforming force. Some substances, such as wax, pitch and glass, which possess the outward appearance of being in the solid state, yield and flow under pressure, and they are sometimes regarded as highly viscous liquids. Solids may be crystalline or amorphous, and the crystalline state differs from the amorphous state in the regular arrangement of the constituent molecules, atoms or ions into some fixed and rigid pattern known as a lattice. Actually, many of the substances that were once considered to be amorphous have now been shown, by X-ray analysis, to exhibit some degree of regular molecular arrangement, but the term `crystalline' is most frequently used to indicate a high degree of internal regularity, resulting in the development of definite external crystal faces. As molecular motion in a gas or liquid is free and random, the physical properties of these fluids are the same no matter in what direction they are measured. In other words, they are isotropic. True amorphous solids, because of the random arrangement of their constituent molecules, are also isotropic. Most crystals, however, are anisotropic; their mechanical, electrical, magnetic and optical properties can vary according to the direction in which they are measured. Crystals belonging to the cubic system are the exception to this rule; their highly symmetrical internal arrangement renders them optically isotropic. Anisotropy is most readily detected by refractive index measurements, and the striking phenomenon of double refraction exhibited by a clear crystal of Iceland spar (calcite) is probably the best-known example.

1.1 Liquid crystals Before considering the type of crystal with which everyone is familiar, namely the solid crystalline body, it is worth while mentioning a state of matter which possesses the flow properties of a liquid yet exhibits some of the properties of the crystalline state.

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2

Crystallization

Although liquids are usually isotropic, some 200 cases are known of substances that exhibit anisotropy in the liquid state at temperatures just above their melting point. These liquids bear the unfortunate, but popular, name `liquid crystals': the term is inapt because the word `crystal' implies the existence of a precise space lattice. Lattice formation is not possible in the liquid state, but some form of molecular orientation can occur with certain types of molecules under certain conditions. Accordingly, the name `anisotropic liquid' is preferred to `liquid crystal'. The name `mesomorphic state' is used to indicate that anisotropic liquids are intermediate between the true liquid and crystalline solid states. Among the better-known examples of anisotropic liquids are p-azoxyphenetole, p-azoxyanisole, cholesteryl benzoate, ammonium oleate and sodium stearate. These substances exhibit a sharp melting point, but they melt to form a turbid liquid. On further heating, the liquid suddenly becomes clear at some fixed temperature. On cooling, the reverse processes occur at the same temperatures as before. It is in the turbid liquid stage that anisotropy is exhibited. The changes in physical state occurring with change in temperature for the case of p-azoxyphenetole are: solid (anisotropic)

137  C

)*

turbid liquid (anisotropic, mesomorphic)

167  C

)*

clear liquid (isotropic)

The simplest representation of the phenomenon is given by Bose's swarm theory, according to which molecules orientate into a number of groups in parallel formation (Figure 1.1). In many respects this is rather similar to the behaviour of a large number of logs floating down a river. Substances that can exist in the mesomorphic state are usually organic compounds, often aromatic, with elongated molecules. The mesomorphic state is conveniently divided into two main classes. The smectic (soap-like) state is characterized by an oily nature, and the flow of such liquids occurs by a gliding movement of thin layers over one another. Liquids in the nematic (thread-like) state flow like normal viscous liquids, but mobile threads can often be observed within the liquid layer. A third class, in which

Figure 1.1. Isotropic and anisotropic liquids. (a) Isotropic: molecules in random arrangement; (b) anisotropic: molecules aligned into swarms

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The crystalline state

3

strong optical activity is exhibited, is known as the cholesteric state; some workers regard this state as a special case of the nematic. The name arises from the fact that cholesteryl compounds form the majority of known examples. For further information on this subject, reference should be made to the relevant references listed in the Bibliography at the end of this chapter.

1.2 Crystalline solids The true solid crystal comprises a rigid lattice of molecules, atoms or ions, the locations of which are characteristic of the substance. The regularity of the internal structure of this solid body results in the crystal having a characteristic shape; smooth surfaces or faces develop as a crystal grows, and the planes of these faces are parallel to atomic planes in the lattice. Very rarely, however, do any two crystals of a given substance look identical; in fact, any two given crystals often look quite different in both size and external shape. In a way this is not very surprising, as many crystals, especially the natural minerals, have grown under different conditions. Few natural crystals have grown `free'; most have grown under some restraint resulting in stunted growth in one direction and exaggerated growth in another. This state of affairs prevented the general classification of crystals for centuries. The first advance in the science of crystallography came in 1669 when Steno observed a unique property of all quartz crystals. He found that the angle between any two given faces on a quartz crystal was constant, irrespective of the relative sizes of these faces. This fact was confirmed later by other workers, and in 1784 HauÈy proposed his Law of Constant Interfacial Angles: the angles between corresponding faces of all crystals of a given substance are constant. The crystals may vary in size, and the development of the various faces (the crystal habit) may differ considerably, but the interfacial angles do not vary; they are characteristic of the substance. It should be noted, however, that substances can often crystallize in more than one structural arrangement (polymorphism ± see section 1.8) in which case HauÈy's law applies only to the crystals of a given polymorph. Interfacial angles on centimetre-sized crystals, e.g. geological specimens, may be measured with a contact goniometer, consisting of an arm pivoted on a protractor (Figure 1.2), but precisions greater than 0.5 are rarely possible. The reflecting goniometer (Figure 1.3) is a more versatile and accurate apparatus. A crystal is mounted at the centre of a graduated turntable, a beam of light from an illuminated slit being reflected from one face of the crystal. The reflection is observed in a telescope and read on the graduated scale. The turntable is then rotated until the reflection from the next face of the crystal is observed in the telescope, and a second reading is taken from the scale. The difference  between the two readings is the angle between the normals to the two faces, and the interfacial angle is therefore (180 ) . Modern techniques of X-ray crystallography enable lattice dimensions and interfacial angles to be measured with high precision on milligram samples of crystal powder specimens.

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4

Crystallization

Figure 1.2. Simple contact goniometer

Figure 1.3. Reflecting goniometer

1.3 Crystal symmetry Many of the geometric shapes that appear in the crystalline state are readily recognized as being to some degree symmetrical, and this fact can be used as a means of crystal classification. The three simple elements of symmetry which can be considered are: 1. Symmetry about a point (a centre of symmetry) 2. Symmetry about a line (an axis of symmetry) 3. Symmetry about a plane (a plane of symmetry) It must be remembered, however, that while some crystals may possess a centre and several different axes and planes of symmetry, others may have no element of symmetry at all. A crystal possesses a centre of symmetry when every point on the surface of the crystal has an identical point on the opposite side of the centre, equidistant from it. A perfect cube is a good example of a body having a centre of symmetry (at its mass centre). If a crystal is rotated through 360 about any given axis, it obviously returns to its original position. If, however, the crystal appears to have reached its original

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The crystalline state

5

Figure 1.4. The 13 axes of symmetry in a cube

Figure 1.5. The 9 planes of symmetry in a cube

position more than once during its complete rotation, the chosen axis is an axis of symmetry. If the crystal has to be rotated through 180 (360/2) before coming into coincidence with its original position, the axis is one of twofold symmetry (called a diad axis). If it has to be rotated through 120 (360/3), 90 (360/4) or 60 (360/6) the axes are of threefold symmetry (triad axis), fourfold symmetry (tetrad axis) and sixfold symmetry (hexad axis), respectively. These are the only axes of symmetry possible in the crystalline state. A cube, for instance, has 13 axes of symmetry: 6 diad axes through opposite edges, 4 triad axes through opposite corners and 3 tetrad axes through opposite faces. One each of these axes of symmetry is shown in Figure 1.4. The third simple type is symmetry about a plane. A plane of symmetry bisects a solid object in such a manner that one half becomes the mirror image of the other half in the given plane. This type of symmetry is quite common and is often the only type exhibited by a crystal. A cube has 9 planes of symmetry: 3 rectangular planes each parallel to two faces, and 6 diagonal planes passing through opposite edges, as shown in Figure 1.5. It can be seen, therefore, that the cube is a highly symmetrical body, as it possesses 23 elements of symmetry (a centre, 9 planes and 13 axes). An octahedron also has the same 23 elements of symmetry; so, despite the difference in outward appearance, there is a definite crystallographic relationship between these two forms. Figure 1.6 indicates the passage from the cubic (hexahedral) to the octahedral form, and vice versa, by a progressive and symmetrical removal of the corners. The intermediate solid forms shown (truncated cube, truncated octahedron and cubo-octahedron) are three of the 13 Archimedean semiregular solids which are called combination forms, i.e. combinations of a cube and an octahedron. Crystals exhibiting combination forms are commonly encountered (see Figure 1.20).

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Figure 1.6. Combination forms of cube and octahedron

The tetrahedron is also related to the cube and octahedron; in fact these three forms belong to the five regular solids of geometry. The other two (the regular dodecahedron and icosahedron) do not occur in the crystalline state. The rhombic dodecahedron, however, is frequently found, particularly in crystals of garnet. Table 1.1 lists the properties of the six regular and semi-regular forms most often encountered in crystals. The Euler relationship is useful for calculating the number of faces, edges and corners of any polyhedron: E ˆF ‡C

2

This relationship states that the number of edges is two less than the sum of the number of faces and corners. A fourth element of symmetry which is exhibited by some crystals is known by the names `compound, or alternating, symmetry', or symmetry about a Table 1.1. Properties of some regular and semi-regular forms found in the crystalline state Form

Regular solids Tetrahedron Hexahedron (cube) Octahedron Semi-regular solids Truncated cube Truncated octahedron Cubo-octahedron

Faces Edges Corners Edges at a corner

Elements of symmetry Centre

Planes

Axes

4 6 8

6 12 12

4 8 6

3 3 4

No Yes Yes

6 9 9

7 13 13

14 14 14

36 36 24

24 24 12

3 3 4

Yes Yes Yes

9 9 9

13 13 13

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Figure 1.7. An axis of compound symmetry

`rotation±reflection axis' or `axis of rotatory inversion'. This type of symmetry obtains when one crystal face can be related to another by performing two operations: (a) rotation about an axis, and (b) reflection in a plane at right angles to the axis, or inversion about the centre. Figure 1.7 illustrates the case of a tetrahedron, where the four faces are marked A, B, C and D. Face A can be transformed into face B after rotation through 90 , followed by an inversion. This procedure can be repeated four times, so the chosen axis is a compound axis of fourfold symmetry.

1.4 Crystal systems There are only 32 possible combinations of the above-mentioned elements of symmetry, including the asymmetric state (no elements of symmetry), and these are called the 32 point groups or classes. All but one or two of these classes have been observed in crystalline bodies. For convenience these 32 classes are grouped into seven systems, which are known by the following names: regular (5 possible classes), tetragonal (7), orthorhombic (3), monoclinic (3), triclinic (2), trigonal (5) and hexagonal (7). The first six of these systems can be described with reference to three axes, x, y and z. The z axis is vertical, and the x axis is directed from front to back and the y axis from right to left, as shown in Figure 1.8a. The angle between the axes y and z is denoted by , that between x and z by , and that between x and y by . Four axes are required to describe the hexagonal system: the z axis is vertical and perpendicular to the other three axes (x, y and u), which are coplanar and inclined at 60 (or 120 ) to one another, as shown in Figure 1.8b. Some workers

Figure 1.8. Crystallographic axes for describing the seven crystal systems: (a) three axes b ˆ ; xz b ˆ ; xy b ˆ ; (b) four axes (hexagonal system) xy ˆ yu b ˆ ux b ˆ 60 (120 ) yz

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Table 1.2. The seven crystal systems System

Other names

Angles between axes

Length of axes

Examples

 ˆ ˆ ˆ 90

xˆyˆz

Sodium chloride Potassium chloride Alums Diamond

 ˆ ˆ ˆ 90

x ˆ y 6ˆ z

Rutile Zircon Nickel sulphate. 7H2 O

 ˆ ˆ ˆ 90

x 6ˆ y 6ˆ z

Potassium permanganate Silver nitrate Iodine -Sulphur

Monoclinic

Monosymmetric  ˆ ˆ 90 6ˆ Clinorhombic Oblique

x 6ˆ y 6ˆ z

Potassium chlorate Sucrose Oxalic acid -Sulphur

Triclinic

Anorthic

 6ˆ 6ˆ 6ˆ 90

x 6ˆ y 6ˆ z

Potassium dichromate Copper sulphate. 5H2 O

xˆyˆz

Sodium nitrate Ruby Sapphire

Regular

Cubic Octahedal Isometric Tesseral

Tetragonal

Pyramidal Quadratic

Orthorhombic Rhombic Prismatic Isoclinic Trimetric

Asymmetric Trigonal

Rhombohedral

 ˆ ˆ 6ˆ 90

Hexagonal

None

z axis is perpen- x ˆ y ˆ u 6ˆ z dicular to the x, y and u axes, which are inclined at 60

Silver iodide Graphite Water (ice) Potassium nitrate

prefer to describe the trigonal system with reference to four axes. Descriptions of the seven crystal systems, together with some of the other names occasionally employed, are given in Table 1.2. For the regular, tetragonal and orthorhombic systems, the three axes x, y and z are mutually perpendicular. The systems differ in the relative lengths of these axes: in the regular system they are all equal; in the orthorhombic system they are all unequal; and in the tetragonal system two are equal and the third is different. The three axes are all unequal in the monoclinic and triclinic systems; in the former, two of the angles are 90 and one angle is different, and in the latter all three angles are unequal and none is equal to 90 . Sometimes the limitation `not equal to 30 , 60 or 90 ' is also applied to the triclinic system. In the trigonal system three equal axes intersect at equal angles, but the angles are

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Figure 1.9. Hemihedral forms of the octahedron and tetragonal bipyramid

not 90 . The hexagonal system is described with reference to four axes. The axis of sixfold symmetry (hexad axis) is usually chosen as the z axis, and the other three equal-length axes, located in a plane at 90 to the z axis, intersect one another at 60 (or 120 ). Each crystal system contains several classes that exhibit only a partial symmetry; for instance, only one-half or one-quarter of the maximum number of faces permitted by the symmetry may have been developed. The holohedral class is that which has the maximum number of similar faces, i.e. possesses the highest degree of symmetry. In the hemihedral class only half this number of faces have been developed, and in the tetrahedral class only one-quarter have been developed. For example, the regular tetrahedron (4 faces) is the hemihedral form of the holohedral octahedron (8 faces) and the wedge-shaped sphenoid is the hemihedral form of the tetragonal bipyramid (Figure 1.9). It has been mentioned above that crystals exhibiting combination forms are often encountered. The simplest forms of any crystal system are the prism and the pyramid. The cube, for instance, is the prism form of the regular system and the octahedron is the pyramidal form, and some combinations of these two forms have been indicated in Figure 1.6. Two simple combination forms in the tetragonal system are shown in Figure 1.10. Figures 1.10a and b are the tetragonal prism and bipyramid, respectively. Figure 1.10c shows a tetragonal prism that is terminated by two tetragonal pyramids, and Figure 1.10d the

Figure 1.10. Simple combination forms in the tetragonal system: (a) tetragonal prism; (b) tetragonal bipyramid; (c) combination of prism and bipyramid; (d ) combination of two bipyramids

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combination of two different tetragonal bipyramids. It frequently happens that a crystal develops a group of faces which intersect to form a series of parallel edges: such a set of faces is said to constitute a zone. In Figure 1.10b, for instance, the four prism faces make a zone. The crystal system favoured by a substance is to some extent dependent on the atomic or molecular complexity of the substance. More than 80 per cent of the crystalline elements and very simple inorganic compounds belong to the regular and hexagonal systems. As the constituent molecules become more complex, the orthorhombic and monoclinic systems are favoured; about 80 per cent of the known crystalline organic substances and 60 per cent of the natural minerals belong to these systems.

1.5 Miller indices All the faces of a crystal can be described and numbered in terms of their axial intercepts. The axes referred to here are the crystallographic axes (usually three) which are chosen to fit the symmetry; one or more of these axes may be axes of symmetry or parallel to them, but three convenient crystal edges can be used if desired. It is best if the three axes are mutually perpendicular, but this cannot always be arranged. On the other hand, crystals of the hexagonal system are often allotted four axes for indexing purposes. If, for example, three crystallographic axes have been decided upon, a plane that is inclined to all three axes is chosen as the standard or parametral plane. It is sometimes possible to choose one of the crystal faces to act as the parametral plane. The intercepts X, Y and Z of this plane on the axes x, y and z are called parameters a, b and c. The ratios of the parameters a : b and b : c are called the axial ratios, and by convention the values of the parameters are reduced so that the value of b is unity. W. H. Miller suggested, in 1839, that each face of a crystal could be represented by the indices h, k and l, defined by hˆ

a b , kˆ X Y

and

lˆ

c Z

Figure 1.11. Intercepts of planes on the crystallographic axes

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For the parametral plane, the axial intercepts X, Y and Z are the parameters a, b and c, so the indices h, k and l are a/a, b/b and c/c, i.e. 1, 1 and 1. This is usually written (111). The indices for the other faces of the crystal are calculated from the values of their respective intercepts X, Y and Z, and these intercepts can always be represented by ma, nb and pc, where m, n and p are small whole numbers or infinity (HauÈy's Law of Rational Intercepts). The procedure for allotting face indices is indicated in Figure 1.11, where equal divisions are made on the x, y and z axes. The parametral plane ABC, with axial intercepts of OA ˆ a, OB ˆ b and OC ˆ c, respectively, is indexed (111), as described above. Plane DEF has axial intercepts X ˆ OD ˆ 2a, Y ˆ OE ˆ 3b and Z ˆ OF ˆ 3c; so the indices for this face can be calculated as 1 2 1 k ˆ b/Y ˆ b/3b ˆ 3 1 l ˆ c/Z ˆ c/3c ˆ 3 h ˆ a/X ˆ a/2a ˆ

Hence h : k : l ˆ 12 : 13 : 13, and multiplying through by six, h : k : l ˆ 3 : 2 : 2. Face DEF, therefore, is indexed (322). Similarly, face DFG, which has axial intercepts of X ˆ 2a, Y ˆ 2b and Z ˆ 3c, gives h : k : l ˆ 12 : 12 : 13 ˆ 3 : 3 : 2 or (3 32). Thus the Miller indices of a face are inversely proportional to its axial intercepts. The generally accepted notation for Miller indices is that (hkl) represents a crystal face or lattice plane, while fhklg represents a crystallographic form comprising all faces that can be derived from hkl by symmetry operations of the crystal. Figure 1.12 shows two simple crystals belonging to the regular system. As there is no inclined face in the cube, no face can be chosen as the parametral plane (111). The intercepts Y and Z of face A on the axes y and z are at infinity,

Figure 1.12. Two simple crystals belonging to the regular system, showing the use of Miller indices

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so the indices h, k and l for this face will be a/a, b/1 and c/1, or (100). Similarly, faces B and C are designated (010) and (001), respectively. For the octahedron, face A is chosen arbitrarily as the parametral plane, so it is designated (111). As the crystal belongs to the regular system, the axial intercepts made by the other faces are all equal in magnitude, but not in sign, to the parametral intercepts a, b and c. For instance, the intercepts of face B on the z axis is negative, so this face is designated (111). Similarly, face C is designated (1 11), and the unmarked D face is (111). Figure 1.13 shows some geometrical figures representing the seven crystal systems, and Figure 1.14 indicates a few characteristic forms exhibited by crystals of some common substances. Occasionally, after careful goniometric measurement, crystals may be found to exhibit plane surfaces which appear to be crystallographic planes, being symmetrical in accordance with the symmetry of the crystal, but which cannot be described by simple indices. These are called vicinal faces. A simple method for determining the existence of these faces is to observe the reflection of a spot of light on the face: four spot reflections, for example, would indicate four vicinal faces. The number of vicinal faces corresponds to the symmetry of the face, and this property may often be used as an aid to the classification of the crystal. For example, a cube face (fourfold axis of symmetry) may appear to be made up of an extremely flat four-sided pyramid with its base being the true (100) plane but its apex need not necessarily be at the centre of the face. An octahedral face (threefold symmetry) may show a three-sided pyramid. These vicinal faces most probably arise from the mode of layer growth on the individual faces commencing at point sources (see section 6.1).

Figure 1.13. The seven crystal systems

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Figure 1.14. Some characteristic crystal forms

1.6 Space lattices The external development of smooth faces on a crystal arises from some regularity in the internal arrangement of the constituent ions, atoms or molecules. Any account of the crystalline state, therefore, should include some reference to the internal structure of crystals. It is beyond the scope of this book to deal in any detail with this large topic, but a brief description will be given of the concept of the space lattice. For further information reference should be made to the specialized works listed in the Bibliography. It is well known that some crystals can be split by cleavage into smaller crystals which bear a distinct resemblance in shape to the parent body. While there is clearly a mechanical limit to the number of times that this process can be repeated, eighteenth century investigators, Hooke and HauÈy in particular, were led to the conclusion that all crystals are built up from a large number of minute units, each shaped like the larger crystal. This hypothesis constituted a very important step forward in the science of crystallography because its logical extension led to the modern concept of the space lattice. A space lattice is a regular arrangement of points in three dimensions, each point representing a structural unit, e.g. an ion, atom or a molecule. The whole

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Table 1.3. The fourteen Bravais lattices Type of symmetry

Lattice

Corresponding crystal system

Cubic

Cube Body-centred cube Face-centred cube

Regular

Tetragonal

Square prism Body-centred square prism

Tetragonal

Orthorhombic

Rectangular prism Body-centred rectangular prism Rhombic prism Body-centred rhombic prism

Orthorhombic

Monoclinic

Monoclinic parallelepiped Clinorhombic prism

Monoclinic

Triclinic

Triclinic parallelepiped

Triclinic

Rhomboidal

Rhombohedron

Trigonal

Hexagonal

Hexagonal prism

Hexagonal

structure is homogeneous, i.e. every point in the lattice has an environment identical with every other point's. For instance, if a line is drawn between any two points, it will, when produced in both directions, pass through other points in the lattice whose spacing is identical with that of the chosen pair. Another way in which this homogeneity can be visualized is to imagine an observer located within the structure; he would get the same view of his surroundings from any of the points in the lattice. By geometrical reasoning, Bravais postulated in 1848 that there were only 14 possible basic types of lattice that could give the above environmental identity. These 14 lattices can be classified into seven groups based on their symmetry, which correspond to the seven crystal systems listed in Table 1.2. The 14 Bravais lattices are given in Table 1.3. The three cubic lattices are illustrated in Figure 1.15; the first comprises eight structural units arranged at the corners of a cube, the second consists of a cubic structure with a ninth unit located at the centre of the cube, and the third of a cube with six extra units each located on a face of the cube.

Figure 1.15. The three cubic lattices: (a) cube; (b) body-centred cube; (c) face-centred cube

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The points in any lattice can be arranged to lie on a larger number of different planes, called lattice planes, some of which will contain more points per unit area than others. The external faces of a crystal are parallel to lattice planes, and the most commonly occurring faces will be those which correspond to planes containing a high density of points, usually referred to as a high reticular density (Law of Bravais). Cleavage also occurs along lattice planes. Bravais suggested that the surface energies, and hence the rates of growth, should be inversely proportional to the reticular densities, so that the planes of highest density will grow at the slowest rate and the low-density planes, by their high growth rate, may soon disappear. For these reasons, the shape of a grown crystal may not always reflect the symmetry expected from its basic unit cell (see section 6.4). Although there are only 14 basic lattices, interpenetration of lattices can occur in actual crystals, and it has been deduced that 230 combinations are posible which still result in the identity of environment of any given point. These combinations are the 230 space groups, which are divided into the 32 point groups, or classes, mentioned above in connection with the seven crystal systems. The law of Bravais has been extended by Donnay and Harker in 1937 into a more generalized form (the Bravais±Donnay±Harker Principle) by consideration of the space groups rather than the lattice types.

1.7 Solid state bonding Four main types of crystalline solid may be specified according to the method of bonding in the solid state, viz. ionic, covalent, molecular and metallic. There are materials intermediate between these classes, but most crystalline solids can be classified as predominantly one of the basic types. The ionic crystals (e.g. sodium chloride) are composed of charged ions held in place in the lattice by electrostatic forces, and separated from the oppositely charged ions by regions of negligible electron density. In covalent crystals (e.g. diamond) the constituent atoms do not carry effective charges; they are connected by a framework of covalent bonds, the atoms sharing their outer electrons. Molecular crystals (e.g. organic compounds) are composed of discrete molecules held together by weak attractive forces (e.g. -bonds or hydrogen bonds). Metallic crystals (e.g. copper) comprise ordered arrays of identical cations. The constituent atoms share their outer electrons, but these are so loosely held that they are free to move through the crystal lattice and confer `metallic' properties on the solid. For example, ionic, covalent and molecular crystals are essentially non-conductors of electricity, because the electrons are all locked into fixed quantum states. Metals are good conductors because of the presence of mobile electrons. Semiconducting crystals (e.g. germanium) are usually covalent solids with some ionic characteristics, although a few molecular solids (e.g. some polycyclic aromatic hydrocarbons such as anthracene) are known in which under certain conditions a small fraction of the valency electrons are free to move in the

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crystal. The electrical conductivity of semiconductors is electronic in nature, but it differs from that in metals. Metallic conductivity decreases when the temperature is raised, because thermal agitation exerts an impeding effect. On the other hand, the conductivity of a semiconductor increases with heating, because the number of electron±`hole' pairs, the electricity carriers in semiconductors, increases greatly with temperature. Metals have electrical resistivities in the ranges 10 8 to 10 6 S 1 m. Insulators cover the range 108 to 1020 (diamond) and semiconductors 10 to 107 S 1 m. The electrical conductivity of a semiconductor can be profoundly affected by the presence of impurities. For example, if x silicon atoms in the lattice of a silicon crystal are replaced by x phosphorus atoms, the lattice will gain x electrons and a negative (n-type) semiconductor results. On the other hand, if x silicon atoms are replaced by x boron atoms, the lattice will lose x electrons and a positive (p-type) semiconductor is formed. The impurity atoms are called `donors' or `acceptors' according to whether they give or take electrons to or from the lattice.

1.8 Isomorphs and polymorphs Two or more substances that crystallize in almost identical forms are said to be isomorphous (Greek: `of equal form'). This is not a contradiction of HauÈy's law, because these crystals do show small, but quite definite, differences in their respective interfacial angles. Isomorphs are often chemically similar and can then be represented by similar chemical formulae; this statement is one form of Mitscherlich's Law of Isomorphism, which is now recognized only as a broad generalization. One group of compounds which obey and illustrate Mitscherlich's law is represented by the formula M20 SO4
M200 (SO4 )3
24H2 O (the alums), where M0 represents a univalent radical (e.g. K or NH4 ) and M 000 represents a tervalent radical (e.g. Al, Cr or Fe). Many phosphates and arsenates, sulphates and selenates are also isomorphous. Sometimes isomorphous substances can crystallize together out of a solution to form `mixed crystals' or, as they are better termed, crystalline `solid solutions'. In such cases the composition of the homogeneous solid phase that is deposited follows no fixed pattern; it depends largely on the relative concentrations and solubilities of the substances in the original solvent. For instance, chrome alum, K2 SO4
Cr2 (SO4 )3
24H2 O (purple), and potash alum, K2 SO4
Al2 (SO4 )3
24H2 O (colourless), crystallize from their respective aqueous solutions as regular octahedra. When an aqueous solution containing both salts is crystallized, regular octahedra are again formed, but the colour of the crystals (which are now homogeneous solid solutions) can vary from almost colourless to deep purple, depending on the proportions of the two alums in the crystallizing solution. Another phenomenon often shown by isomorphs is the formation of overgrowth crystals. For example, if a crystal of chrome alum (octahedral) is placed in a saturated solution of potash alum, it will grow in a regular manner such that the purple core is covered with a continuous colourless overgrowth. In

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a similar manner an overgrowth crystal of nickel sulphate, NiSO4
7H2 O (green), and zinc sulphate, ZnSO4
7H2 O (colourless), can be prepared. There have been many `rules' and `tests' proposed for the phenomenon of isomorphism, but in view of the large number of known exceptions to these it is now recognized that the only general property of isomorphism is that crystals of the different substances shall show very close similarity. All the other properties, including those mentioned above, are merely confirmatory and not necessarily shown by all isomorphs. A substance capable of crystallizing into different, but chemically identical, crystalline forms is said to exhibit polymorphism. Different polymorphs of a given substance are chemically identical but will exhibit different physical properties. Dimorphous and trimorphous substances are commonly known, e.g. Calcium carbonate: calcite (trigonal-rhombohedral) aragonite (orthorhombic) vaterite (hexagonal) Carbon: graphite (hexagonal) diamond (regular) Silicon dioxide: cristobalite (regular) tridymite (hexagonal) quartz (trigonal) The term allotropy instead of polymorphism is often used when the substance is an element. The different crystalline forms exhibited by one substance may result from a variation in the crystallization temperature or a change of solvent. Sulphur, for instance, crystallizes in the form of orthorhombic crystals (-S) from a carbon disulphide solution, and of monoclinic crystals ( -S) from the melt. In this particular case the two crystalline forms are interconvertible: -sulphur cooled below 95.5  C changes to the  form. This interconversion between two crystal forms at a definite transition temperature is called enantiotropy (Greek: `change into opposite') and is accompanied by a change in volume. Ammonium nitrate (melting point 169.6  C) exhibits five polymorphs and four enantiotropic changes between 18 and 125  C, as shown below: liquid

…I†

* ) 169:6  C

cubic

* ) 125:2  C

) * 32:3 C

orthorhombic

…IV†

…II†

trigonal

)

* 18 C 

) * 84:2 C

…III†

orthorhombic

…V†

tetragonal

The transitions from forms II to III and IV to V result in volume increases: the changes from I to II and III to IV are accompanied by a decrease in volume. These volume changes frequently cause difficulty in the processing and storage of ammonium nitrate. The salt can readily burst a metal container into which it has been cast when change II to III occurs. The drying of ammonium nitrate crystals must be carried out within fixed temperature limits, e.g. 40±80  C,

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otherwise the crystals can disintegrate when a transition temperature is reached. Crystals of polymorphic substances sometimes undergo transformation without a change of external form, the result being an aggregate of very small crystals of the stable modification confined within the boundaries of the original unstable form. For example, an unstable rhombohedral form of potassium nitrate can crystallize from a warm aqueous solution, but when these crystals come into contact with a crystal of the stable modification, transformation sweeps rapidly through the rhombohedra which retain their shape. The crystals lose much of their transparency and acquire a finely granular appearance and their original mechanical strength is greatly reduced. Such pseudomorphs as they are called exhibit confused optical properties which cannot be correlated with the external symmetry (Hartshorne and Stuart, 1969). When polymorphs are not interconvertible, the crystal forms are said to be monotropic: graphite and diamond are monotropic forms of carbon. The term isopolymorphism is used when each of the polymorphous forms of one substance is isomorphous with the respective polymorphous form of another substance. For instance, the regular and orthorhombic polymorphs of arsenious oxide, As2 O3 , are respectively isomorphous with the regular and orthorhombic polymorphs of antimony trioxide, Sb2 O3 . These two oxides are thus said to be isodimorphous. Polytypism is a form of polymorphism in which the crystal lattice arrangements differ only in the manner in which identical two-dimensional arrays are stacked (Verma and Krishna, 1966).

1.9 Enantiomorphs and chirality Isomeric substances, different compounds having the same formula, may be divided into two main groups: (a) constitutional isomers, which differ because their constituent atoms are connected in a different order, e.g., ethanol CH3 CH2 OH and dimethylether CH3 OCH3 , (b) stereoisomers, which differ only in the spacial arrangement of their constituent atoms. Stereoisomers can also be divided into two groups: (i) enantiomers, molecules that are mirror images of one another, and (ii) diastereomers, which are not. Diastereomers can have quite different properties, e.g., the cis- and transcompounds maleic and fumaric acids which have different melting points, 130  C and 270  C respectively. On the other hand, enantiomers have identical properties with one exception, viz., that of optical activity, the ability to rotate the plane of polarization of plane-polarized light. One form will rotate to the right (dextrorotatory) and the other to the left (laevorotatory). The direction and magnitude of rotation are measured with a polarimeter. Molecules and substances that exhibit optical activity are generally described as chiral (Greek cheir `hand'). Two crystals of the same substance that are

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19

mirror images of each other are said to be enantiomorphous (Greek `of opposite form'). These crystals have neither planes of symmetry nor a centre of symmetry. Enantiomorphous crystals are not necessarily optically active, but all known optically active substances are capable of being crystallized into enantiomorphous forms. In many cases the solution or melt of an optically active crystal is also optically active. However, if dissolution or melting destroys the optical activity, this is an indication that the molecular structure was not enantiomeric. Tartaric acid (Figure 1.16) and certain sugars are well-known examples of optically active substances. Optical activity is generally associated with compounds that possess one or more atoms around which different elements or groups are arranged asymmetrically, i.e., a stereocentre, so that the molecule can exist in mirror image forms. The most common stereocentre in organic compounds is an asymmetric carbon atom, and tartaric acid offers a good example. Three possible arrangements of the tartaric acid molecule are shown in Figure 1.17. The (a) and (b) forms are mirror images of each other; both contain asymmetric carbon atoms and both are optically active; one will be the dextro-form and the other the laevo-form. Although there are two asymmetric carbon atoms in formula (c), this particular form (meso-tartaric acid) is optically inactive; the potential optical activity of one-half of the molecule is compensated by the opposite potential optical activity of the other. Dextro- and laevo-forms are now designated in all modern texts as (‡) and ( ) respectively. The optically inactive racemate, a true double compound

Figure 1.16. (a) Dextro- and (b) laevo-tartaric acid crystals (monoclinic system)

Figure 1.17. The tartaric acid molecule: (a) and (b) optically active forms; (c) mesotartaric acid, optically inactive

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Crystallization

(section 4.3.2), comprising an equimolar mixture of (‡) and ( ) forms, is designated (). The symbols d- and l-, commonly found in older literature to designate optically active dextro- and laevo-forms, were abandoned to avoid confusion with the capital letters D and L which are still commonly used to designate molecular configuration, but not the direction of rotation of planepolarized light. It is important to note, therefore, that not all D series compounds are necessarily dextrorotatory (‡) nor are all L series compounds laevorotatory ( ). The D , L system, was arbitrarily based on the configuration of the enantiomeric glyceraldehyde molecules: the (‡)-isomer was taken to have the structure implied by formula 1 and this arrangement of atoms was called the D configuration. Conversely, formula 2 was designated as representing the L configuration:

CHO

CHO H

C

OH

CH2OH (1)

HO

H

C

CH2OH (2)

Lactic acid provides a simple example of how the D , L system could be applied to other compounds. The relative configuration of lactic acid is determined by the fact that it can be synthesized from (D )-(‡)-glyceraldehyde without breaking any bonds to the asymmetric carbon atom:

COOH

CHO H

C

OH

CH 2 OH (D)-(+)-glyceraldehyde

H

C

OH

CH 3 (D)-(–)-lacticacid

The lactic acid produced by this reaction, however, is laevorotatory not dextrorotatory like the starting material, thus illustrating the above warning that there is no essential link between optical rotation and molecular configuration. The D , L system becomes ambiguous for all but the simplest of molecules and is now increasingly being replaced with the more logically based and adaptable R , S system, which has been internationally adopted by IUPAC for classifying absolute molecular configuration. Comprehensive accounts of the R , S convention and its application are given in most modern textbooks on organic chemistry, but the following short introduction may serve as a brief guide to the procedure for classifying a compound with a single asymmetric carbon atom as the stereocentre.

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21

First, the four different groups attached to the stereocentre are identified and each is assigned a priority number, 1 to 4, using the Cahn±Ingold±Prelog `sequence rules' according to which the highest priority (1) is given to the group with the atom directly attached to the stereocentre that has the highest atomic number. If by this rule two or more groups would at first appear to have identical priorities, the atomic numbers of the second atoms in each group are compared, continuing with the subsequent atoms until a difference is identified. For example, for -aminopropionic acid (alanine)

CH3 H

C

NH2

COOH the following priorities would be assigned: NH2 ˆ 1, COOH ˆ 2, CH3 ˆ 3 and H ˆ 4. The model of the molecule is then oriented in space so that the stereocentre is observed from the side opposite the lowest priority group. So observing the stereocentre with the lowest priority group (H ˆ 4) to the rear, the view would be (1) NH2

either

(1) NH2

or R

L C

C H3C

COOH

(3)

(2)

HOOC (2)

CH 3 (3)

According to the Cahn±Ingold±Prelog rules, if the path from 1 to 2 to 3 runs clockwise the stereocentre is designated by the letter R (Latin: rectus, right). If the path runs anticlockwise it is designated by the letter S (Latin: sinister, left). If the structure has only one stereocentre, (R ) or (S ) is used as the first prefix to the name, e.g., (S )-aminopropionic acid. The optical rotation of the compound is indicated by a second prefix, e.g., (S )-(‡)-aminopropionic acid, noting again as mentioned above for the D , L system, there is no necessary connection between (S ) left and (R ) right configurations and the ( ) left and (‡) right directions of optical rotation. If the molecule has more than one stereocentre their designations and positions are identified in the prefix, e.g., (2R , 3R )dibromopentane. 1.9.1 Racemism The case of tartaric acid serves to illustrate the property known as racemism. An equimolar mixture of crystalline D and L tartaric acids dissolved in water will produce an optically inactive solution. Crystallization of this solution will

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yield crystals of optically inactive racemic acid which are different in form from the D and L crystals. There is, however, a difference between a racemate and a meso-form of a substance; the former can be resolved into D and L forms but the latter cannot. Crystalline racemates are normally considered to belong to one of two basic classes: 1. Conglomerate: an equimolal mechanical mixture of two pure enantiomorphs. 2. Racemic compound: an equimolal mixture of two enantiomers homogeneously distributed throughout the crystal lattice. A racemate can be resolved in a number of ways. In 1848 Pasteur found that crystals of the sodium ammonium tartrate (racemate) Na
NH4
C4 H4 O6
H2 O deposited from aqueous solution, consisted of two clearly different types, one being the mirror image of the other. The D and L forms were easily separated by hand picking. Although widely quoted, however, this example of manual resolution through visual observation is in fact a very rare occurrence. Bacterial attack was also shown by Pasteur to be effective in the resolution of racemic acid. Penicillium glaucum allowed to grow in a dilute solution of sodium ammonium racemate destroys the D form but, apart from being a rather wasteful process, the attack is not always completely selective. A racemate may also be resolved by forming a salt or ester with an optically active base (usually an amine) or alcohol. For example, a racemate of an acidic substance A with, say, the dextro form of an optically active base B will give DLA

‡ DB ! DA
DB ‡ LA
DB

and the two salts DA
DB and LA
DB can then be separated by fractional crystallization. A comprehensive account of the resolution of racemates is given by Jacques, Collet and Wilen (1981). This topic is further discussed in section 7.2.

1.10 Crystal habit Although crystals can be classified according to the seven general systems (Table 1.1), the relative sizes of the faces of a particular crystal can vary considerably. This variation is called a modification of habit. The crystals may grow more rapidly, or be stunted, in one direction; thus an elongated growth of the prismatic habit gives a needle-shaped crystal (acicular habit) and a stunted growth gives a flat plate-like crystal (tabular, platy or flaky habit). Nearly all manufactured and natural crystals are distorted to some degree, and this fact frequently leads to a misunderstanding of the term `symmetry'. Perfect geometric symmetry is rarely observed in crystals, but crystallographic symmetry is readily detected by means of a goniometer.

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Figure 1.18. Crystal habit illustrated on a hexagonal crystal

Figure 1.19. Some common habits of potassium sulphate crystals (orthorhombic system): a ˆ f100g, b ˆ f010g, c ˆ f011g, l ˆ f021g, m ˆ f110g, o ˆ f111g, t ˆ f130g

Figure 1.18 shows three different habits of a crystal belonging to the hexagonal system. The centre diagram (b) shows a crystal with a predominant prismatic habit. This combination-form crystal is terminated by hexagonal pyramids and two flat faces perpendicular to the vertical axis; these flat parallel faces cutting one axis are called pinacoids. A stunted growth in the vertical direction (or elongated growth in the directions of the other axes) results in a tabular crystal (a); excessively flattened crystals are usually called plates or flakes. An elongated growth in the vertical direction yields a needle or acicular crystal (c); flattened needle crystals are often called blades. Figure 1.19 shows some of the habits exhibited by potassium sulphate crystals grown from aqueous solution and Figure 1.20 shows four different habits of sodium chloride crystals. The relative growths of the faces of a crystal can be altered, and often controlled, by a number of factors. Rapid crystallization, such as that produced by the sudden cooling or seeding of a supersaturated solution, may result in the formation of needle crystals; impurities in the crystallizing solution can stunt

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Crystallization

Figure 1.20. Four different habits of sodium chloride (regular system) crystals (Courtesy of ICI Ltd, Mond Division)

the growth of a crystal in certain directions; and crystallization from solutions of the given substance in different solvents generally results in a change of habit. The degree of supersaturation or supercooling of a solution or melt often exerts a considerable influence on the crystal habit, and so can the state of agitation of the system. These and other factors affecting the control of crystal habit are discussed in section 6.4.

1.11 Dendrites Rapid crystallization from supercooled melts, supersaturated solutions and vapours frequently produces tree-like formations called dendrites, the growth of which is indicated in Figure 1.21. The main crystal stem grows quite rapidly in a supercooled system that has been seeded, and at a later stage primary branches grow at a slower rate out of the stem, often at right angles to it. In certain cases, small secondary branches may grow slowly out of the primaries. Eventually branching ceases and the pattern becomes filled in with crystalline material. Most metals crystallize from the molten state in this manner, but because of the filling-in process the final crystalline mass may show little outward appear-

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25

Figure 1.21. Dendritic growth

ance of dendrite formation. The fascinating patterns of snow crystals are good examples of dendritic growth, and the frosting of windows often affords a visual observation of this phenomenon occurring in two dimensions. The growth of a dendrite can be observed quite easily under a microscope by seeding a drop of a supersaturated solution on the slide. Dendrites form most commonly during the early stages of crystallization; at later stages a more normal uniform growth takes place and the pattern may be obliterated. Dendritic growth occurs quite readily in thin liquid layers, probably because of the high rate of evaporative cooling, whereas agitation tends to suppress this type of growth. Dendrite formation tends to be favoured by substances that have a high enthalpy of crystallization and a low thermal conductivity.

1.12 Composite crystals and twins Most crystalline natural minerals, and many crystals produced industrially, exhibit some form of aggregation or intergrowth, and prevention of the formation of these composite crystals is one of the problems of large-scale crystallization. The presence of aggregates in a crystalline mass spoils the appearance of the product and interferes with its free-flowing nature. More important, however, aggregation is often indicative of impurity because crystal clusters readily retain impure mother liquor and resist efficient washing (section 9.7.2). Composite crystals may occur in simple symmetrical forms or in random clusters. The simplest form of aggregate results from the phenomenon known as parallel growth; individual forms of the same substance grow on the top of one another in such a manner that all corresponding faces and edges of the individuals are parallel. Potash alum, K2 SO4
Al2 (SO4 )3
24H2 O, exhibits this type of growth; Figure 1.22 shows a typical structure in which regular octahedra

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Crystallization

Figure 1.22. Parallel growth on a crystal of potash alum

Figure 1.23. Interpenetrant twin of two cubes (e.g. fluorspar)

are piled on top of one another in a column symmetrical about the vertical axis. Parallel growth is often associated with isomorphs; for instance, parallel growths of one alum can be formed on the crystals of another, but this property is no longer regarded as an infallible test for isomorphism. Another composite crystal frequently encountered is known as a twin or a macle; it appears to be composed of two intergrown individuals, similar in form, joined symmetrically about an axis (a twin axis) or a plane (a twin plane). A twin axis is a possible crystal edge and a twin plane is a possible crystal face. Many types of twins may be formed in simple shapes such as a V, ‡, L and so forth, or they may show an interpenetration giving the appearance of one individual having passed completely through the other (Figure 1.23). Partial interpenetration (Figure 1.24) can also occur. In some cases, a twin crystal may present the outward appearance of a form that possesses a higher degree of symmetry than that of the individuals, and this is known as mimetic twinning. A typical example of this behaviour is orthorhombic potassium sulphate, which can form a twin looking almost identical with a hexagonal bipyramid. Parallel growth and twinning (or even triplet formation) are usually encountered when crystallization has been allowed to take place in an undisturbed medium. Although twins of individuals belonging to most of the seven crystal systems are known, twinning occurs most frequently when the crystals belong to the orthorhombic or monoclinic systems. Certain impurities in the crystallizing medium can cause twin formation even under vigorously agitated conditions: this is one of the problems encountered in the commercial crystallization of sugar. The formation of crystal clusters, aggregates or conglomerates which possess no symmetrical properties is probably more frequently encountered in largescale crystallization than the formation of twins. Relatively little is still known

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Figure 1.24. Partial interpenetrant twin (e.g. quartz)

about the growth of these irregular crystal masses, but among the factors that generally favour their formation are poor agitation, the presence of certain impurities in the crystallizing solution, seeding at high degrees of supersaturation and the presence of too many seed crystals, leading to conditions of overcrowding in the crystallizer.

1.13 Imperfections in crystals Very few crystals are perfect. Indeed, in many cases they are not required to be, since lattice imperfections and other defects can confer some important chemical and mechanical properties on crystalline materials. Surface defects can also greatly influence the process of crystal growth. There are three main types of lattice imperfection: point (zero-dimensional, line (one-dimensional) and surface (two-dimensional). 1.13.1 Point defects The common point defects are indicated in Figure 1.25. Vacancies are lattice sites from which units are missing, leaving `holes' in the structure. These units may be atoms, e.g. in metallic crystals, molecules (molecular crystals) or ions (ionic crystals). The interstitials are foreign atoms that occupy positions in the interstices between the matrix atoms of the crystal. In most cases the occurrence of interstitials leads to a distortion of the lattice. More complex point defects can occur in ionic crystals. For example, a cation can leave its site and become relocated interstitially near a neighbouring cation. This combination of defects (a cation vacancy and an interstitial cation) is called a Frenkel imperfection. A cation vacancy combined with an anion vacancy is called a Schottky imperfection. A foreign atom that occupies the site of a matrix atom is called a substitutional impurity. Many types of semiconductor crystals contain controlled quantities of substitutional impurities. Germanium crystals, for example, can be grown containing minute quantities of aluminium (p-type semiconductors) or phosphorus (n-type).

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Crystallization

Figure 1.25. Representation of some common point defects: A, interstitial impurity; B, substitutional impurity; C, vacancy

1.13.2 Line defects The two main types of line defect which can play an important role in the model of crystal growth are the edge and screw dislocations. Both of these are responsible for slip or shearing in crystals. Large numbers of dislocations occur in most crystals; they form readily during the growth process under the influence of surface and internal stresses. Figure 1.26 shows in diagrammatic form the cross-sectional view of a crystal lattice in which the lower part of a vertical row of atoms is missing. The position of the dislocation is marked by the symbol ?; the vertical stroke of this symbol indicates the extra plane of atoms and the horizontal stroke indicates the slip plane. The line passing through all the points ?, i.e. drawn vertical to the plane of the diagram, is called the edge dislocation line. In an edge dislocation, therefore, the atoms are displaced at right angles to the dislocation line. The process of slip under the action of a shearing force may be explained as follows (see Figure 1.26). The application of a shear stress to a crystal causes atom A to move further away from atom B and closer to atom C. The bond between A and B, which is already strained, breaks and a new bond is formed between A and C. The dislocation thus moves one atomic distance to the right, and if this process is continued the dislocation will eventually reach the edge of the crystal. The direction and magnitude of slip are indicated by the Burgers vector, which may be one or more atomic spacings. In the above example, where the displacement is one lattice spacing, the Burgers vector is equal to 1.

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Figure 1.26. Movement of an edge dislocation through a crystal

Figure 1.27. A screw dislocation

A screw dislocation forms when the atoms are displaced along the dislocation line, rather than at right angles to it as in the case of the edge dislocation. Figure 1.27 indicates this type of lattice distortion. In this example the Burgers vector is 1 (unit step height), but its magnitude may be any integral number. Screw dislocations give rise to a particular mode of growth in which the attachment of growth units to the face of the dislocation results in the development of a spiral growth pattern over the crystal face (see section 6.1.2).

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Crystallization

Figure 1.28. A simple tilt boundary

1.13.3 Surface defects A variety of surface imperfections, or mismatch boundaries, can be produced in crystalline materials as a result of mechanical or thermal stresses or irregular growth. Grain boundaries, for example, can be created between individual crystals of different orientation in a polycrystalline aggregate. When the degree of mismatching is small, the boundary can be considered to be composed of a line of dislocations. A low-angle tilt boundary is equivalent to a line of edge dislocations, and the angle of tilt is given by  ˆ b/h where b is the Burgers vector and h the average vertical distance between the dislocations (Figure 1.28). A twist boundary can be considered, when the degree of twist is small, as a succession of parallel screw dislocations. For a full account of this subject reference should be made to the specialized works (see Bibliography).

Bibliography Buckley, H. E. (1952) Crystal Growth. London: Wiley Bunn, C. W. (1961) Chemical Crystallography, 2nd edn. Oxford: Clarendon Press Chandrasekhar, S. (1992) Liquid Crystals, 2nd edn. Cambridge: University Press Hartshorne, N. H. and Stuart, A. (1969) Practical Optical Crystallography, 2nd edn. London: Arnold Hirth, J. P. and Loethe, J. (1968) Theory of Dislocations. New York: McGraw-Hill Kelly, A. and Groves, G. (2000) Crystallography and Crystal Defects. Chichester: Wiley

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McKie, D. and McKie, C. (1974) Crystalline Solids, 3rd edn. London: Nelson Phillips, F. C. (1980) Introduction to Crystallography, 4th edn. London: Longmans Green Rosenberg, H. M. (1978) The Solid State. Oxford: Clarendon Press Vainshtein, B. K., Chernov, A. A. and Shuvalov, L. A. (1984) (eds.) Modern Crystallography, vol. 1 Symmetry of Crystals; vol. 2 Structure of Crystals; vol. 3 Crystal Growth; vol. 4 Physical Properties of Crystals. Berlin: Springer Van Bueren, H. G. (1960) Imperfections in Crystals. New York: Interscience Wells, A. F. (1962) Structural Inorganic Chemistry, 3rd edn. Oxford: Clarendon Press Wright, J. D. (1987) Molecular Crystals. Cambridge: University Press

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Physical and thermal properties

2.1 Density 2.1.1 Solids The densities of most pure solid substances are readily available in the standard physical property handbooks. The densities of actual crystallized substances, however, may differ from the literature values on account of the presence of vapour or liquid inclusions (see section 6.6) or adhering surface moisture. For example, recorded values of `commercially pure' sucrose crystals have ranged from 1580 to 1610 kg m 3 compared with an expected value of 1587 kg m 3 . The theoretical density, c , of a crystal may be calculated from the lattice parameters (section 1.6) by means of the relationship: nM (2:1) VN Where n is the number of formula units in the unit cell, V is the volume of the unit cell, M is the molar mass of the substance and N is the Avogadro number (6:023  1026 kmol 1 ). For sucrose, the lengths of the a, b and c axes of this monoclinic crystal are 10.9, 8.70 and 7:75  10 10 m, respectively, with the angle ˆ 103 (sin ˆ 0:9744). Hence the volume V ˆ 716:1  10 30 m3 . Further, M ˆ 342:3 kg kmol 1 and n ˆ 2. Substituting these values in equation 2.1 gives a value of c ˆ 1587 kg m 3 . The actual density of a solid substance, even of a relatively small individual crystal, may be measured by determining the density of an inert liquid mixture in which the crystal remains just suspended. Examples of a convenient group of miscible organic liquids for many inorganic salts include chloroform (1492 kg m 3 at 20  C), carbon tetrachloride (1594), ethyl iodide (1930), ethylene dibromide (2180), bromoform (2890) and methylene iodide (3325). For example, a crystal of sodium nitrate (2260 kg m 3 ) could be floated in about 50 mL of bromoform in a suitable flask, taking care that no air bubbles are attached, and then caused to achieve the `just suspended' state by slowly adding chloroform from a burette. At this point the crystal density may be assumed to be equal to that of the liquid mixture, which can readily be estimated. Solid densities have a very small temperature dependence, but this can be ignored for industrial crystallization purposes. For example the density of sodium chloride decreases by about 0.7 per cent when the temperature increases from 10 to 80  C. The calculation needs a knowledge of the coefficient of thermal expansion. The densities of bulk particulate solids and slurries are discussed in section 2.1.3. c ˆ

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2.1.2 Liquids The density of a liquid is significantly temperature dependent. The ratio of the density of a given liquid at one temperature to the density of water at the same, or another, temperature is known as the specific gravity of the liquid. Thus, a specific gravity quoted at say 20  C/4  C is numerically equal to the density of the liquid at 20  C expressed in g cm 3 since water exhibits its maximum density (1 g cm 3 ) at 4  C. The simplest instrument for measuring liquid density is the hydrometer, a float with a graduated stem. To approach reasonable accuracy, however, it is essential to make the measurement at the particular calibration temperature marked on the hydrometer. Densities may be determined more accurately by the specific gravity bottle method, or with a pyknometer (BS 733, 1983) or Westphal balance, details of which may be found in most textbooks of practical physics. In recent years, several high-precision instruments have become available, the most noteworthy of which are those based on an oscillating sample holder. A glass U-tube is filled with the sample and caused to oscillate at its natural frequency, which is dependent on the total mass of the system. Since the tube has a constant mass and sample volume, the measured frequency of oscillation can be related to the liquid sample density. Precisions of up to 10 6 g cm 3 have been claimed for some instruments. It is often possible to estimate to 5% the density of a solution from a knowledge of the solute and solvent densities by means of the equation soln ˆ

L‡S L S ‡ L S

(2:2)

where L and S are the masses of the solvent (liquid) and solute (solid), respectively, and L and S are the densities of the respective components. It has to be acknowledged, of course, that the volume of a solution is not exactly equal to the volumes of the solvent and added solute, but the error incurred in making this assumption is often insignificant, particularly for industrial purposes, as the following examples show: 1. An aqueous solution of potassium sulphate at 80  C containing 0:214 kg K2 SO4 /kg water (L ˆ 971:8 kg m 3 at 80  C, S ˆ 2660 kg m 3 ). soln ˆ (1 ‡ 0:214)/[(1/971:8) ‡ (0:214/2660)] ˆ 1095 kg m 3 : Experimental value: 1117 kg m 3 : 2. An aqueous solution of sodium sulphate at 15  C containing 0:429 kg Na2 SO4
10H2 O/kg `free' water (L ˆ 999:1 kg m 3 at 15  C, S ˆ 1460 kg m 3 ). soln ˆ (1 ‡ 0:429)/[(1/999:1) ‡ (0:429/1460)] ˆ 1103 kg m Experimental value: 1125 kg m 3 :

3

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Crystallization

3. An aqueous solution of sucrose at 80  C containing 3:62 kg C6 H12 O6 /kg water (L ˆ 971:8 kg m 3 at 80  C, S ˆ 1590 kg m 3 ). soln ˆ (1 ‡ 3:62)/[(1/971:8) ‡ (3:62/1590)] ˆ 1397 kg m Experimental value: 1350 kg m 3 :

3

For more reliable estimations of solution density, reference should be made to the procedures described in the book by SoÈhnel and Novotny (1985) which also contains detailed density data for a large number of inorganic salt solutions. In a later publication (Novotny and SoÈhnel, 1988) densities of some 300 inorganic salt solutions are recorded. The densities of some aqueous solutions are recorded in the Appendix (Table A.9). 2.1.3 Bulk solids and slurries The bulk density of a quantity of particulate solids is not a fixed property of the system since the bulk volume occupied contains significant amounts of void space, normally filled with air. The relationship between the density of the solid particles, S , and the bulk solids density, BS , is BS ˆ S (1

")

(2:3)

where " is the voidage, the volume fraction of voids, which is considerably dependent on particle shape, particle size distribution and the packing of the particles. Further, " can vary considerably depending on how the particulate material has been processed or handled. For example, it can be increased by aeration, as in freshly poured solids, and decreased by vibration, e.g. after transportation in packaged form. Expression of slurry densities Many different terms are used for specifying the solids content of slurries and each has its own particular use. Slurry concentrations are not usually simple to measure experimentally since it is not always convenient to filter-off, wash, dry and weigh the solids content. So other more convenient, but less precise methods, are often adopted. For example, it is common practice to take a sample of slurry in a graduated cylinder, allow the solids to settle and to measure the volume percentage of settled solids. Although this is often a rapid and quite satisfactory way of assessing the slurry concentration, particularly for routine testing under industrial plant conditions, it does not directly give the actual quantity of suspended solids, because the settled volume (the overall volume occupied by the settled solids) contains a significant proportion of liquid. Settled spheres of uniform size, for example, enclose a void space of about 40 per cent, but considerable deviations from this value can occur for multisized particles of irregular shape.

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The relationship between system voidages and the settled solids fraction is given by equation 2.4. Other useful relationships are given in equations 2.5 to 2.9. "ˆ1 ˆ1

S(1 "S ) (MT /S )

ˆ (  S )/(S MT ˆ (1 ")S ˆ X  ˆ X[S where

(2:4) (2:5) L )

(2:6) (2:7) (2:8)

"(S

L )]

(2:9)

S ˆ settled solids fraction (m3 of settled solids plus associated liquor in the voids/m3 of total sample taken) X ˆ mass fraction of solids (kg of suspended solids/kg of total suspension) MT ˆ slurry density (kg of suspended solids/m3 of total suspension) " ˆ voidage of the slurry (m3 of liquid/m3 of total suspension) 1 " ˆ volume fraction of solids (m3 of solids/m3 of total suspension) "S ˆ voidage of the settled solids  ˆ overall mean density of the suspension (kg m 3) S ˆ density of solid (kg m 3) L ˆ density of liquid (kg m 3)

For the special case of an industrial crystallizer, it is sometimes possible to assess the slurry (magma) density by chemical analysis, measuring (a) the total overall concentration of solute (crystals plus dissolved solute) in the suspension and (b) the concentration of dissolved solute in the supernatant liquor. Thus, C ˆ MT ‡ "C 

(2:10)

and MT ˆ S (C

C  )/(S

C )

(2:11)

where C ˆ kg of crystallizing substance (suspended and dissolved)/m3 of total suspension, and C  ˆ kg of dissolved crystallizing substance /m3 of supernatant liquor.

2.2 Viscosity The once common units of absolute viscosity, the poise (P) (1 g cm 1 s 1 ) and its useful sub-multiple the centipoise (cP), have now been replaced by the SI unit (kg m 1 s 1 ) which is generally written as Pa s and sometimes as N s m 2 . The following relationships hold: 1 cP ˆ 0:01P ˆ 1 mPa s ˆ 1 mN s m ˆ 2:42 lb ft

1

h

1

2

ˆ 10

3

kg m

1

s

1

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Similarly, the once common units of kinematic viscosity (ˆ absolute viscosity/ density), the stokes (St) (1 cm2 s 1 ) and its sub-multiple the centistokes (cSt), have been replaced by the SI unit (m2 s 1 ). The following relationships hold: 1 cSt ˆ 0:01 St ˆ 10

6

m2 s

1

ˆ 0:0388 ft2 h

1

The viscosity of a liquid decreases with increasing temperature, and for many liquids the relationship  ˆ A exp( B/T)

(2:12)

holds reasonably well. A and B are constants and the temperature T is expressed in kelvins. Plots of log  versus T 1 or log  versus log T usually yield fairly straight lines and this property may be used for interpolating viscosities at temperatures within the range covered. In general, dissolved solids increase the viscosity of water, although a few exceptions to this rule are known. Occasionally, the increase in viscosity is considerable, as in the case of the system sucrose±water where, for example, the viscosity increases from around 2 to 60 mPa s for a concentration increase from 20 to 60 g/100 g of solution at 20  C. Figure 2.1a shows an example of a solute that decreases the viscosity of the solvent; in this system (KI±water) a minimum viscosity is exhibited. Several other potassium and ammonium salts also exhibit a similar behaviour. Figure 2.1b shows the effect of concentration and temperature on the ethanol±water system which exhibits a maximum viscosity. A comprehensive survey of the viscosity characteristics of aqueous solutions of electrolytes has been made by Stokes and Mills (1965) who also give experimental data on a considerable number of systems. Viscosities of some aqueous solutions are recorded in the Appendix (Table A.10). Unfortunately, no completely reliable method is available for the prediction of the viscosities of solutions or liquid mixtures. A general survey is made by

Figure 2.1. Aqueous solutions exhibiting (a) minimum, (b) maximum viscosities. (After Hatschek, 1928)

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Figure 2.2. Simple U-tube viscometer

Reid, Prausnitz and Poling (1987) and some indication of the complexities involved in making estimates of the viscosities of mixed salt solutions may be gained from the survey made by Nowlan, Thi and Sangster (1980). Numerous instruments have been devised for the measurement of liquid viscosity, many of which are based on the flow of a fluid through a capillary tube and the application of Poiseuille's law in the form ˆ

r4
P 8lV

(2:13)

where
P ˆ the pressure drop across the capillary of length l and radius r, and V ˆ volume of fluid flowing in unit time. One simple type of U-tube viscometer is shown in Figure 2.2. The liquid under test is sucked into leg B until the level in this leg reaches mark z. The tube is arranged truly vertical, and the temperature of the liquid is measured and kept constant. The liquid is then sucked up into leg A to a point above x and the time t for the meniscus to fall from x to y is recorded. The kinematic viscosity of the liquid v can be calculated from v ˆ kt

(2:14)

where k is a constant for the apparatus, determined by measurements on a liquid of known viscosity, e.g. water. The falling-sphere method of viscosity determination also has many applications, and Stokes' law may be applied in the form ˆ

(S

1 )d 2 g 18u

(2:15)

where d,  and u are the diameter, density and terminal velocity, respectively, of a solid sphere falling in the liquid of density 1 . A simple falling-sphere viscometer is shown in Figure 2.3. The liquid under test is contained in the inner

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Figure 2.3. Falling-sphere viscometer

tube of 30 mm diameter and about 300 mm length. The central portion of the tube contains two reference marks a and b, 150 mm apart. The tube is held truly vertical, and the temperature of the liquid is measured and kept constant. A 1.5 mm diameter steel ball, previously warmed to the test temperature, is inserted through a small guide tube and its time of passage between the two reference marks is measured. A mean of several measurements should be taken. The viscosity of the liquid can then be calculated from equation 2.15. Modern instrumental versions of the falling-sphere technique are claimed to measure viscosities in the range 0.5±500 mPa with high precision at controlled temperatures using sample volumes as low as 0.5 mL. Several high-precision viscometers are based on the concentric-cylinder method. The liquid under test is contained in the annulus between two vertical coaxial cylinders; one cylinder can be made to rotate at a constant speed, and the couple required to prevent the other cylinder rotating can be measured. For more detailed information on practical viscometry reference should be made to specialized publications (Dinsdale and Moore 1962, BS 188, 1993). 2.2.1 Solid±liquid systems The viscosity characteristics of liquids can be altered considerably by the presence of finely dispersed solid particles, especially of colloidal size. The viscosity of a suspension of rigid spherical particles in a liquid, when the distance between the spheres is much greater than their diameter, may be expressed by the Einstein equation: S ˆ 0 (1 ‡ 2:5)

(2:16)

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where S is the effective viscosity of the disperse system, 0 the viscosity of the pure dispersion medium, and  the ratio of the volume of the dispersed particles to the total volume of the disperse system. In other words,  ˆ 1 ", where " is the voidage of the system. Equation 2.16 applies reasonably well to lyophobic sols and very dilute suspensions, but for moderately concentrated suspensions and lyophilic sols the Guth±Simha modification is preferred: S ˆ 0 (1 ‡ 2:5 ‡ 14:12 )

(2:17)

For concentrated suspensions of solid particles, the Frankel±Acrivos (1967) relationship ! 9 (/m )1/3 S ˆ  0 (2:18) 8 (1 /m )1/3 can be applied for values of /m ! 1, where m is the maximum attainable volumetric concentration of solids in the system (usually about 0.6 for packed monosize spherical particles). Equation 2.18 has met with experimental support in the region (/m ) > 0:7. However, for solids concentrations such as those normally encountered in industrial crystallizers (say "  0:8,   0:2 and /m  0:3 for granular crystals) the much simpler equation 2.17 predicts the order of magnitude of apparent viscosity reasonably well.

2.3 Surface tension Of the many methods available for measuring the surface tension of liquids (Findlay, 1973), the capillary rise and ring techniques are probably the most useful for general applications. In the capillary rise method, the surface tension, , of a liquid can be determined from the height, h, of the liquid column in a capillary tube of radius r. If the liquid completely wets the tube (zero contact angle),

ˆ 12 rh
g

(2:19)

where
 is the difference in density between the liquid and the gaseous atmosphere above it. The height, h, can be accurately measured with a cathetometer from the base of the liquid meniscus to the flat surface of the free liquid surface in a containing vessel. However, to minimize errors, this reference to a flat surface can be eliminated by measuring the difference in capillary rise in two tubes of different bore (Figure 2.4). Then

ˆ 12 r1 h1
g ˆ 12 r2 h2
g From which it follows that

ˆ


h
r1 r2 g 2(r1 r2 )

The differential height
h can be measured with precision.

(2:20)

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Figure 2.4. Measurement of surface tension by the differential capillary rise method

The ring technique, and its many variations, is widely used in industrial laboratories. Several kinds of commerical apparatus incorporating a torsion balance are available under the name du NouÈy tensometer. The method is simple and rapid, and is capable of measuring the surface tension of a pure liquid to a precision of 0.3% or better. The force necessary to pull a ring (usually of platinum or platinum±iridium wire) from the surface of the liquid is measured. The surface tension is calculated from the pull and the dimensions of the ring after the appropriate correction factors have been applied. It is often possible to predict the surface tension of non-aqueous mixtures of solvents by assuming a linear dependence with mole fraction. Aqueous solutions, however, generally show a pronounced non-linear behaviour and prediction is not recommended. The surface tension of a liquid decreases with an increase in temperature, but the decrease is not always linear (Table 2.1). The addition of an electrolyte to water generally increases the surface tension very slightly, although an initial decrease is usually observed at very low concentrations (< 0:002 mol L 1 ) (Harned and Owen, 1958). Non-electrolytes generally decrease the surface tension of water. For example, saturated aqueous solutions of -naphthol, adipic acid and benzoic acid at 22  C are 48, 55 and 60 mN m 1 , respectively, whereas a saturated solution of potassium sulphate at the same temperature has a surface tension of 73 mN m 1 .

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Table 2.1. Surface tensions of some common solvents at different temperatures Solvent Water Benzene Toluene CCl4 Acetone Methanol Ethanol 1 mN m

1

ˆ 1 mJ m

0

10

76.0 31.6 30.8 29.5 25.5 24.3 24.1

73.5 30.2 29.7 28.0 24.4 23.4 23.1

2

1

ˆ 1 dyn cm

Surface tension, mN m 20 30 72.8 28.9 28.5 26.8 23.3 22.6 22.3

71.2 27.2 27.4 25.5 22.3 21.7 21.4

1

40

50  C

69.6 26.3 26.2 24.4 21.2 20.8 20.6

67.9 25.0 25.1 23.1 ± ± 19.8

2.3.1 Interfacial tension The surface tension of a liquid, as normally measured, is the interfacial tension between a liquid surface and air saturated with the relevant vapour. The interfacial tension of a crystalline solid in contact with a solution of the dissolved solid is a quantity of considerable importance in crystal nucleation and growth processes. It is also sometimes referred to as the `surface energy'. This subject is dealt with in section 5.6.

2.4 Diffusivity Two examples of a theoretical approach to the problem of the prediction of diffusion coefficients in fluid media are the equations postulated in 1905 by Einstein and in 1936 by Eyring. The former is based on kinetic theory and a modification of Stokes' law for the movement of a particle in a fluid, and is most conveniently expressed in the form Dˆ

kT r

(2:21)

where D ˆ diffusivity (m2 s 1 ), T ˆ absolute temperature (K),  ˆ viscosity (kg s 1 m 1 ), r ˆ molecular radius (m), k ˆ Boltzmann's constant and the dimensionless factor  has a numerical value between 4 and 6 depending on the solute : solvent molecular size ratio. Eyring's approach, based on reaction rate theory, treats a liquid as a disordered lattice structure with vacant sites into which molecules move, i.e. diffuse. For low solute concentration Eyring's equation may be expressed in a form identical with that of equation 2.21 but a different value of  applies. The usefulness of these equations, however, is strictly limited because they both contain a term, r, which denotes the radius of the solute molecule. Values of this quantity are difficult to obtain. Consequently, the most directly useful relationship that emerges is

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D/T ˆ constant

(2:22)

which is of considerable value in predicting, for a given system, the effect of temperature and viscosity on the diffusion coefficient. This simple relation is often referred to as the Stokes±Einstein equation. The very limited success of the theoretical equations has led to the development of many empirical and semi-empirical relationships for the prediction of diffusion coefficients. Amongst those devised for diffusion in liquids the following may be mentioned. For diffusion in aqueous solutions Othmer and Thakar (1953) proposed the correlation Dˆ

14  10 1:1 v0:6

5

(2:23)

where v is the molar volume of the solute (cm3 mol 1 ) and  is the viscosity in cP, giving the diffusivity D in cm2 s 1 . By correlating a large number of published experimental diffusivities Wilke and Chang (1955) arrived at the relationship D ˆ 7:4  10

8

( M)T 1/2 v0:6

(2:24)

where M is the molar mass (kg kmol 1 ), v is the molal volume (cm3 mol 1 ) and T is in kelvins, giving D in cm2 s 1 . For unassociated solvents, e.g. benzene, ether and heptane, the so-called association parameter ˆ 1. For water, methanol and ethanol, ˆ 2:6, 1:9 and 1.5, respectively. Despite the widespread use of these and many other similar correlations, however, they are notoriously unreliable; deviations from experimental values as high as 30% are not unusual (Mullin and Cook, 1965). Furthermore, these empirical relationships were devised from diffusion data predominantly on liquid±liquid systems, and there is little evidence to suggest that they are reliable for the prediction of the diffusion of solid solutes in liquid solutions, although an `order of magnitude' estimation is sometimes possible. For example, the diffusivity of sodium chloride in water at 25  C is 1:3  10 9 m2 s 1 , while values calculated from equations 2.23 and 2.24 range from 1.7 to 2:6  10 9 m2 s 1 . Similarly, for sucrose in water at 25  C, D ˆ 5:2 10 10 m2 s 1 , while the predicted values range from 3.6 to 4:2  10 10 m2 s 1 . It is also important to note that these empirical correlations are meant to apply only to dilute solutions. Despite the fact that they all contain a term relating to viscosity which is a function of concentration, they usually fail to predict the rate of diffusion from a concentrated solution to a less concentrated one. For example, the diffusion coefficient for sucrose diffusing from a 1% aqueous solution into water at 25  C is approximately five times the value for the diffusion between 61.5 and 60:5% solutions, whereas over the concentration range 1 to 60%, the viscosity exhibits a fortyfold increase. These empirical equations also fail to discriminate between isomers as was pointed out by Mullin and Cook (1965), who measured the diffusivities of o-, m- and p-hydroxybenzoic acid in water. The data measured at 20  C are

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Figure 2.5. Diffusivities of saturated aqueous solutions of the hydroxybenzoic acids into water: * ˆ ortho-, & ˆ meta-, 4= para-, = predicted from equation 2.24 (after Mullin and Cook, 1965)

compared with values predicted by equation 2.24 in Figure 2.5, where the deviation is about 30% between the estimated values and those measured for the m- and p-isomers, with a difference of about 60% between the o- and m-. Clearly equation 2.24 and other empirical relationships, fail to take some property of the system into account, and it is likely that this quantity is the `size' of the diffusing component. For the case of the hydroxybenzoic acids the differences in diffusivity can be accounted for by considering the different hydrogen bonding tendencies of the three isomers, which in turn would influence both the size and shape of the diffusing species.

2.4.1 Experimental measurements In a diffusion cell, where two liquids are brought into contact at a sharp boundary, three different states of diffusion may be recognized. In the case of `free' diffusion, concentrations change progressively away from the interface; when concentrations begin to change at the ends of the cell, `restricted' diffusion is said to occur. If the concentration at a given point in the cell remains constant with respect to time, `steady-state' diffusion is taking place, and, as in all other steady-state processes, a constant supply of material to and removal from the system is required. Several comprehensive accounts have been given of the methods used for measuring diffusion coefficients under these three conditions (Tyrrell, 1961; Stokes and Mills, 1965; Robinson and Stokes, 1970).

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The techniques used for restricted and steady-state diffusion generally involve the use of a diaphragm cell. This method has the disadvantage that the cell has first to be calibrated with a system with a known diffusion coefficient, and for systems with relatively slow diffusivities each run may require an inconveniently long time. The diffusion coefficient under free-diffusion conditions can be measured by analysis at the termination of the experiment or by continuous or intermittent analysis while diffusion continues. Care has to be taken not to disturb the system, and the most widely employed methods for measuring diffusion coefficients in liquids are interferometric, resulting from the original work of Gouy in the 19th century. Two kinds of diffusivity can be recorded, viz. the differential, D, and the integral, D. The differential diffusivity is a value for one particular concentration, c, and driving force, c1 c2 , where c ˆ (c1 ‡ c2 )/2 and c1 c2 is sufficiently small for D to remain unchanged over the concentration range. However, the diffusion coefficient is usually concentration dependent, and in most cases it is the integral diffusivity that is normally measured. This is an average value over the concentration range c1 to c2 . The differential diffusivity is of considerable theoretical importance, and it is only through this quantity that experimental measurements by different techniques can be compared. On the other hand, it is the integral coefficient that is generally required for mass transfer assessment, since this coefficient represents the true `average' diffusivity over the concentration range involved in the mass transfer process. For example, in the dissolution of a solid into a liquid, the solute diffuses from the saturated solution at the interface to the bulk solution. In crystallization the solute diffuses from the supersaturated bulk solution to the saturated solution at the interface. The relevant diffusivity that should be used in an analysis of these two processes, therefore, is the integral diffusivity, which covers the range of concentration from equilibrium saturation to that in the bulk solution. Integral diffusivities may be measured directly by the diaphragm cell technique (Dullien and Shemilt, 1961), but unless the concentration on both sides of the diaphragm (see Figure 2.6) is maintained constant throughout the experiment, the diffusivity measured is the rather complex double-average known as d , defined by an integrated form of the `diaphragm cell integral diffusivity'. D Fick's law of diffusion:   d ˆ 1 ln c10 c20 D (2:25) t c1 t c2 t where c0 and ct are the initial and final (at time t) concentrations, 1 and 2 refer to the lower and upper cells, and is the cell constant (m 2 ), which can be calculated from the cell dimensions:   A 1 1 ˆ ‡ (2:26) L V1 V2

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Figure 2.6. Diaphragm diffusion cell (after Dullien and Shemilt, 1961). A, light liquid compartment; B, heavy liquid compartment; C, stop-cock; D, E, capillaries; F, sintered glass diaphragm; G ˆ polythene-coated iron stirrers; H, rotating magnets

where A and L ˆ area and thickness of diaphragm, and V1 and V2 ˆ volume of the cell compartments. The above method is extremely time-consuming and the process of converting integral to differential values is both tedious and inaccurate. It is more practicable, therefore, to measure the differential diffusivity, D, at intervals over the whole concentration range and to calculate the required integral  by means of the relationship diffusivity, D, Z c1 ˆ 1 D Ddc (2:27) c1 c2 c2 Figure 2.7 shows the measured differential and calculated integral diffusivities for the systems KCl H2 O and NH4 Cl H2 O (Nienow, Unahabhokha and

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Figure 2.7. Measured differential (.) and calculated integral (broken line) diffusivities for the systems NH4 Cl H2 O and KCl H2 O at 20  C (after Nienow, Unahabhokha and Mullin, 1968)

Mullin, 1968). Diffusivities for a number of common electrolyte systems are given in the Appendix (Table A.11). The diffusivity of a strong electrolyte at infinite dilution is called the Nernst limiting value of the diffusion coefficient, D0 , which can be calculated from D0 ˆ

RT(1 2 ) 01 02
01 ‡ 02 F2 1 jz1 j

(2:28)

where 01 and 02 are the limiting conductivities, and 1 and 2 are the number of cations and anions of valency z1 and z2 , respectively. Using the conditions of neutrality: 1 z 1 ‡ 2 z 2 ˆ 0 so equation 2.28 may also be written D0 ˆ

RT jz1 j ‡ jz2 j 01 02

jz1 z2 j 01 ‡ 02 F2

(2:29)

where F is the Faraday constant (9:6487  104 C mol 1 ), R is the gas constant (8:3143 J K 1 mol 1 ), giving values of RT/F2 of 2.4381, 2.6166 and 2:7951  10 2 S 1 mol 1 s 1 at 0, 20 and 40  C, respectively. These may be used with the values of 0 (S m2 mol 1 ) in the Appendix (Table A.13) to give values 1 of D0 in m2 s . By the application of reaction rate theory to both viscosity and diffusion it can be shown that  ˆ A exp( EV /RT)

(2:30)

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and D ˆ B exp( ED /RT)

(2:31)

A plot of log  against 1/T should yield a straight line, and similarly a linear relationship should exist between log D and 1/T. The slopes of these plots give the respective energies of activation EV (viscosity) and ED (diffusion).

2.5 Refractive index Refractometric measurements can often be used for the rapid measurement of solution concentration. Several standard instruments (AbbeÂ, Pulfrich, etc.) are available commercially. A sodium lamp source is most usually used for illumination, and an instrument reading to the fourth decimal place is normally adequate for crystallization work. It is advisable that calibration curves be measured, in terms of temperature and concentration, prior to the study with the actual system. If a dipping-type refractometer is used, a semi-continuous measurement may be made of the change in concentration as the system crystallizes. However, if nucleation is heavy or if large numbers of crystals are present, it may be difficult to provide sufficient illumination for the prism because of the light scattering. One solution to this problem (Leci and Mullin, 1968) is to use a fibre optic (a light wire) fitted into a collar around the prism illuminated from an external source (Figure 2.8). In this way undue heating of the solution is also avoided.

Figure 2.8. A technique for illuminating the prism of a dipping-type refractometer in an opaque solution: A, Perspex collar; B, fibre optic holder; C, fibre optic; D, refractometer prism; E, polished face of prism

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2.6 Electrolytic conductivity The electrolytic conductance of dilute aqueous solutions can often be measured with high precision and thus afford a useful means of determining concentration (see section 3.6.3). A detailed account of the methods used in this area is given by Robinson and Stokes (1970). In the author's experience, however, conductivity measurements are of limited use in crystallization work because of the unreliability of measurement in near-saturated or supersaturated solutions. The temperature dependence of electrical conductivity usually demands a very high precision of temperature control. Torgesen and Horton (1963) successfully operated conductance cells for the control of ADP crystallization, but they had to control the temperature to 0:002  C.

2.7 Crystal hardness Crystals vary in hardness not only from substance to substance but also from face to face on a given crystal (Brookes, O'Neill and Redfern, 1971). One of the standard tests for hardness in non-metallic compounds and minerals is the scratch test, which gave rise to the Mohs scale. Ten `degrees' of hardness are designated by common minerals in such an order that a given mineral will scratch the surface of any of the preceding members of the scale (see Table 2.2). The hardness of metals is generally expressed in terms of their resistance to indentation. A hard indenter is pressed into the surface under the influence of a known load and the size of the resulting indentation is measured. A widely used instrument is the Vickers indenter, which gives a pyramidal indentation, and the results are expressed as a Vickers hardness number (kgf mm 2 ). Other

Table 2.2. Mohs scale of hardness Mohs hardness number M

Reference substance

Formula

Vickers hardness number V

1 2 3 4 5 6 7 8 9 10

talc gypsum calcite fluorite apatite orthoclase quartz topaz corundum diamond

3MgO
4SiO2
H2 O CaSO4
2H2 O CaCO3 CaF2 CaF2
3Ca3 (PO4 )2 K2 O
Al2 O3
6SiO2 SiO2 (AlF)2
SiO4 Al2 O3 C

50 80 130 200 320 500 800 1 300 2 000 10 000

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tests include the Rockwell, which uses a conical indenter and the Brinell, which uses a hard steel ball (Bowden and Tabor, 1964). The relation between Mohs hardness, M, and the Vickers hardness, V, is not a clear one. However, if diamond (M ˆ 10) is omitted from the Mohs scale, the relationship log V ˆ 0:2 M ‡ 1:5

(2:32)

may be used for rough approximation purposes for values of M < 9. A few typical surface hardnesses (Mohs) of some common substances are: sodium potassium lead magnesium

0:5 0:5 1:5 2

aluminium gold brass glass

2 3 2:5 3 3 4 3 4

The scratch test is not really suitable for specifying the hardness of substances commonly crystallized from aqueous solutions, because their Mohs values lie in a very short range, frequently between 1 and 3 for inorganic salts and below 1 for organic substances. For a reliable measurement of hardness of these soft crystals the indentation test is preferred. Ridgway (1970) has measured mean values of the Vickers hardness for several crystalline substances: sodium thiosulphate potassium alum ammonium alum potassium dihydrogen phosphate sucrose

(Na2 S2 O3
5H2 O) (KAl(SO4 )2
12H2 O) (NH4 Al(SO4 )2
12H2 O) (KH2 PO4 ) (C12 H22 O11 )

: 18 : 56 : 58 : 150 : 64

He has also determined the hardnesses of different faces of the same crystal: ammonium dihydrogen phosphate potassium sulphate

(NH4 H2 PO4 ) (100) (110) (K2 SO4 ) (100) (110) (210)

: 69 : 73 : 95 : 100 : 130

Hardness appears to be closely related to density (proportional to) and to atomic or molecular volume (inversely proportional), but few reliable data are available. In a recent study, Ulrich and Kruse (1989) made some interesting comments on these relationships and confirmed the need for more experimental data before any acceptable prediction method can be developed.

2.8 Units of heat The SI heat energy unit is the joule (J), but four other units still commonly encountered are the calorie (cal), kilocalorie (kcal), British thermal unit (Btu) and the centigrade heat unit (chu). The old definitions of these four units are

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Table 2.3. Equivalent values of the common heat energy units cal

kcal

1 1000 252 453.6 0.238 8

0.001 1 0.252 0.453 6 2:388  10

Btu

4

chu

0.003 97 3.97 1 1.8 9:478  10

0.002 21 2.21 0.556 1 5:275  10

4

J

4

4.187 4187 1.055 1.898 1

based on the heat energy required to raise the temperature of a unit mass of water by one degree: 1 cal ˆ 1 g of water raised through 1  C (or K) 1 kcal ˆ 1 kg of water raised through 1  C (or K) 1 chu ˆ 1 lb of water raised through 1  C (or K) 1 Btu ˆ 1 lb of water raised through 1  F The definitions of these units are linked to the basic SI unit, the joule: 1 J ˆ 1 Ws ˆ 1 N m Table 2.3 indicates the equivalent values of these various heat units.

2.9 Heat capacity The amount of heat energy associated with a given temperature change in a given system is a function of the chemical and physical states of the system. A measure of this heat energy can be quantified in terms of the quantity known as the heat capacity which may be expressed on a mass or molar basis. The former is designated the specific heat capacity (J kg 1 K 1 ) and the latter the molar heat capacity (J mol 1 K 1 ). The relationships between some commonly used heat capacity units are: specific heat capacity, C 1 cal g

1

C

1

(or K 1 ) ˆ 1 Btu lb

1

F

1

ˆ 1 chu lb

1

C

1

ˆ 4:187 kJ kg

1

(or K 1 )

K

1

molar heat capacity, C 1 cal mol

1

C (or K 1 ) ˆ 1 Btu lb-mol

1

F

1

ˆ 1 chu lb-mol

1

C

1

K

1

ˆ 4:187 J mol

1

(or K 1 )

For gases two heat capacities have to be considered, at constant pressure, Cp , and at constant volume, Cv . The value of the ratio of these two quantities, Cp /Cv ˆ , varies from about 1.67 for monatomic gases (e.g. He) to about 1.3

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for triatomic gases (e.g. CO2 ). For liquids and solids there is little difference between Cp and Cv , i.e.  1, and it is usual to find Cp values only quoted in the literature. 2.9.1 Solids Specific heat capacities of solid substances near normal atmospheric temperature can be estimated with a reasonable degree of accuracy by combining two empirical rules. The first of these, due to Dulong and Petit, expresses a term called the `atomic heat' which is defined as the product of the relative atomic mass and the specific heat capacity. For all solid elemental substances, the atomic heat is assumed to be roughly constant: atomic heat ' 6:2 cal mol

1

C

1

which in SI units is equivalent to approximately 26 J mol 1 K 1 . However, since virtually all the data available in the literature are recorded in calorie units, these will be retained in this section for all the examples. The second rule, due to Kopp, applies to solid compounds and may be expressed by molar heat capacity ˆ sum of the atomic heats of the constituent atoms In applying these rules, the following exceptions to the approximation `atomic heat ' 6:2' must be noted: C ˆ 1:8 O ˆ 4:0

H ˆ 2:3 F ˆ 5:0

B ˆ 2:7 S ˆ 5:4

Si ˆ 3:8 [H2 O] ˆ 9:8

The substance [H2 O] refers to water as ice or as water of crystallization in solid substances. Obviously a reliable measured value of a heat capacity is preferable to an estimated value, but in the absence of measured values, Kopp's rule can prove extremely useful. A few calculated and observed values of the molar heat capacity are compared in Table 2.4. Table 2.4. Estimated (Kopp's rule) and observed values of molar heat capacity of several solid substances at room temperature Solid Sodium chloride Magnesium sulphate Iodobenzene Naphthalene Potassium sulphate Oxalic acid

Formula

Calculation

NaCl MgSO4
7H2 O C6 H5 I C10 H8 K2 SO4 C2 H2 O4
2H2 O

6:2 ‡ 6:2 6:2 ‡ 5:4 ‡ 4(4:0) ‡ 7(9:8) 6(1:8) ‡ 5(2:3) ‡ 6:2 10(1:8) ‡ 8(2:3) 2(6:2) ‡ 5:4 ‡ 4(4:0) 2(1:8) ‡ 2(2:3) ‡ 4(4:0) ‡ 2(9:8)

C cal mol Calc.

1

C 1 Obs.

12.4 96.2 28.5 36.4 33.8

12.4 89.5 24.6 37.6 30.6

43.8

43.5

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Many inorganic solids have values of specific heat capacity, cp , in the range 0.1±0.3 cal g 1  C 1 (0:4±1:3 kJ kmol 1 K 1 ) and many organic solids have values in the range 0.2±0.5 (0.8±2.1). In general, the heat capacity increases slightly with an increase in temperature. For example, the values of cp at 0 and 100  C for sodium chloride are 0.21 and 0:22 cal g 1  C 1 (8.8 and 9:2 kJ kmol 1 K 1 ) respectively, and the corresponding values for anthracene are 0.30 and 0.35 (approximately 13 and 15 kJ kmol 1 K 1 ). 2.9.2

Pure liquids

A useful method for estimating the molar heat capacity of an organic liquid is based on the additivity of the heat capacity contributions [C ] of the various atomic groupings in the molecules (Johnson and Huang, 1955). Table 2.5 lists some [C ] values, and the following examples illustrate the use of the method ± the molar capacity values (cal mol 1  C 1 ) in parentheses denote values obtained experimentally at 20  C: methyl alcohol (CH3
OH) 9:9 ‡ 11:0 ˆ 20:9 (19:5) toluene (C6 H5
CH3 ) 30:5 ‡ 9:9 ˆ 40:4 (36:8) CH3 isobutyl acetate CH3
COO
CH CH2
CH3 ˆ 3(9:9) ‡ 14:5 ‡ 5:4 ‡ 6:3 ˆ 55:9 (53:3) The heat capacity of a substance in the liquid state is generally higher than that of a substance in the solid state. A large number of organic liquids have specific heat capacity c values in the range 0.4±0.6 cal g 1  C 1 (1.7±2.5 kJ kg 1 K 1 ) at about room temperature. The heat capacity of a liquid usually increases with increasing temperature: for example, the values of c for ethyl alcohol at 0, 20

Table 2.5. Contributions of various atomic groups to the molar heat capacity (cal mol 1  C 1 ) of organic liquids at 20  C (after A. I. Johnson and C. J. Huang, 1955) Group

[C ]

C6 H5 CH3 CH2 CH COOH COO (esters) CˆˆO (ketones) H (formates)

30.5 9.9 6.3 5.4 19.1 14.5 14.7 3.6

Group OH NO2 NH2 CN Cl Br S O (ethers)

[C ] 11.0 15.3 15.2 13.9 8.6 3.7 10.6 8.4

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and 60  C are 0.54, 0.56 and 0:68 cal g 1  C 1 (2.3, 2.3 and 2:9 kJ kg 1 K 1 ) and those for benzene at 20, 40 and 60  C are 0.40, 0.42 and 0.45 (1.7, 1.8 and 2:1 kJ kg 1 K 1 ). Water is an exceptional case; it has a very high heat capacity and exhibits a minimum value at 30  C. The values for water at 0, 15, 30 and 100  C are 1.008, 1.000, 0.9987 and 1:007 cal g 1  C 1 , respectively.

2.9.3 Liquid mixtures and solutions Although not entirely reliable, the following relationship may be used to predict the molar heat capacity C of a mixture of two or more liquids: Cmixt ˆ xA CA ‡ xB CB ‡




(2:33)

where x denotes the mole fraction of the given component in the mixture. A similar relationship may be used to give a rough estimate of the specific heat capacity c: cmixt ˆ XA cA ‡ XB cB ‡




(2:34)

where X is a mass fraction. For example, the value of c for methanol at 20  C is 2:4 kJ kg 1 K 1 . From equation 2.34 it can be calculated that an aqueous solution containing 75 mass per cent of methanol has a specific heat capacity of 2:9 kJ kg 1 K 1 , which coincides closely with measured values. For dilute aqueous solutions of inorganic salts, a rough estimate of the specific heat capacity can be made by ignoring the heat capacity contribution of the dissolved substance, i.e. jcj ˆ j1

Yj

(2:35)

where Y ˆ mass of solute/mass of water, and c ˆ cal g 1  C 1 . Thus, solutions containing 5 g NaCl, 10 g KCl and 15 g CuSO4 per 100 g of solution would by this method be estimated to have specific heat capacities of 0.95, 0.89 and 0:82 cal g 1 C 1 , respectively. Measured values (25  C) for these solutions are 0.94, 0.91 and 0:83 calg 1 C 1 (3.9, 3.8 and 3:5 kJ kg 1 K 1 ), respectively. This estimation method cannot be applied to aqueous solutions of non-electrolytes or acids. Another rough estimation method for the specific heat capacity of aqueous solutions is based on the empirical relationship jcj ˆ j 1 j

(2:36)

where  ˆ density of the solution in g cm 3 . For example, at 30  C a 2 per cent aqueous solution of sodium carbonate by mass has a density of 1:016 g cm 1 and a specific heat capacity of 0:98 cal g 1  C 1 (1/ ˆ 0:98), while a 20 per cent solution has a density of 1.210 and a specific heat capacity of 0:86 (1/ ˆ 0:83).

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2.9.4 Experimental measurement The specific heat capacity of a liquid may be measured by comparing its cooling rate with that of water. This is most conveniently done in a calorimeter, a copper vessel fitted with a copper lid and heavy copper wire stirrer, supported in a draught-free space kept at constant temperature. The vessel is filled almost to the top with a known mass of water at about 70  C and its temperature is recorded every 2 minutes as it cools to, say, 30  C. Significant errors may be incurred if the starting temperature exceeds about 70  C because evaporation greatly increases the rate of cooling. The water is then replaced by the same volume of the liquid or solution under test and its cooling curve is determined over the same temperature range. Taking the specific heat capacities of copper and water to be 385 and 4185 J kg 1 K 1 respectively, a balance of the heat losses gives (4185mw ‡ 385mc )(d/dt)w ˆ (c1 m1 ‡ 385mc )(d/dt)1 where mc ˆ mass of the copper calorimeter, lid and stirrer, mw ˆ mass of water and m1 ˆ mass of liquid. The cooling rates (d/dt) are taken from the slopes of the cooling curves at the chosen temperature , and the specific heat capacity c1 of the liquid under test can thus be evaluated.

2.10 Thermal conductivity The thermal conductivity, , of a substance is defined as the rate of heat transfer by conduction across a unit area, through a layer of unit thickness, under the influence of a unit temperature difference, the direction of heat transmission being normal to the reference area. Fourier's equation for steady conduction may be written as dq ˆ dt

A

d dx

(2:37)

where q, t, A,  and x are units of heat, time, area, temperature and length (thickness), respectively. The SI unit for thermal conductivity is W m 1 K 1 although other units such as cal s 1 cm 1 C 1 and Btu h 1 ft 1 F 1 are still commonly encountered. The conversion factors are: 1 cal s

1

cm

1

C

1

ˆ 418:7 W m

1

K

1

ˆ 241:9 Btu h

1

ft

1

F

1

The thermal conductivity of a crystalline solid can vary considerably according to the crystallographic direction, but very few directional values are available in the literature. Some overall values (W m 1 K 1 ) for polycrystalline or non-crystalline substances include KCl (9.0), NaCl (7.5), KBr (3.8), NaBr (2.5), MgSO4
7H2 O(2:5), ice (2:2), K2 Cr2 O7 (1:9), borosilicate glass (1.0), soda glass (0.7) and chalk (0.7).

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Table 2.6. Thermal conductivities of some pure liquids Temperature,  C 10 40 95

Water 0.59 0.62 0.67

Thermal conductivity,  (W m 1 K 1 ) Acetone Benzene Methanol Ethanol 0.16 0.15 ±

0.14 0.15 ±

0.21 0.19 ±

0.18 0.17 ±

CCl4 0.11 0.099 ±

An increase in the temperature of a liquid usually results in a slight decrease in thermal conductivity, but water is a notable exception to this generalization. Furthermore, water has a particularly high thermal conductivity compared with other pure liquids (Table 2.6 ). There are relatively few values of thermal conductivity for solutions recorded in the literature and regrettably they are frequently conflicting. There is no generally reliable method of estimation. The thermal conductivity of an aqueous solution of a salt is generally slightly lower than that of pure water at the same temperature. For example, the values at 25  C for water, and saturated solutions of sodium chloride and calcium chloride are 0.60, 0.57 and 0:54 W m 1 K 1 , respectively. The thermal conductivity  (W m 1 K 1 ) of pure liquids between about 0 and 70  C may be roughly estimated by the equation  ˆ 3:6  10 8 c(/M)1/3

(2:38)

where c is the specific heat capacity (J kg 1 K 1 ),  is the density (kg m 3 ) and M is the molar mass (kg kmol 1 ).

2.11 Boiling, freezing and melting points When a non-volatile solute is dissolved in a solvent, the vapour pressure of the solvent is lowered. Consequently, at any given pressure, the boiling point of a solution is higher and the freezing point lower than those of the pure solvent. For dilute ideal solutions, i.e. such as obey Raoult's law, the boiling point elevation and freezing point depression can be calculated by an equation of the form
T ˆ

mK M

(2:39)

where m ˆ mass of solute dissolved in a given mass of pure solvent and M ˆ molar mass of the solute. When
T refers to the freezing point depression, K ˆ Kf , the cryoscopic constant; when
T refers to the boiling point elevation, K ˆ Kb , the ebullioscopic constant. Values of Kf and Kb for several common solvents are given in Table 2.7; these, in effect, give the depression in freezing point, or elevation in boiling point, in  C when 1 mol of solute is dissolved, without dissociation or association, in 1 kg of solvent.

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Table 2.7. Cryoscopic and ebullioscopic constants for some common solvents Solvent

Freezing point,  C

Acetic acid Acetone Aniline Benzene Carbon disulphide Carbon tetrachloride Chloroform Cyclohexane Nitrobenzene Methyl alcohol Phenol Water

16.7 95.5 6.2 5.5 108.5 22.6 63.5 6.2 5.7 97.8 42.0 0.0

Boiling point,  C 118.1 56.3 184.5 80.1 46.3 76.8 61.2 80 211 64.7 181 100.0

Kf K kg mol 3.9 2.7 5.9 5.1 3.8 32.0 4.8 20.0 7.0 2.6 7.3 1.86

1

Kb K kg mol

1

3.1 1.7 3.2 2.7 2.4 4.9 3.8 2.8 5.3 0.8 3.0 0.52

The cryoscopic and ebullioscopic constants can be calculated from values of the enthalpies of fusion and vaporization, respectively, by the equation Kˆ

RT 2
H

(2:40)

When K ˆ Kf , T refers to the freezing point Tf (K) and
H to the enthalpy of fusion,
Hf (J kg 1 ). When K ˆ Kb , T refers to the boiling point Tb and
H ˆ
Hvb , the enthalpy of vaporization at the boiling point. The gas constant R ˆ 8:314 J mol 1 K 1 . Boiling points and freezing points are both frequently used as criteria for the estimation of the purity of near-pure liquids. Detailed specifications of standard methods for their determination are given, for example, in the British Pharmacopoeia (2000). Equation 2.40 cannot be applied to concentrated solutions or to aqueous solutions of electrolytes. In these cases the freezing point depression cannot readily be estimated. The boiling point elevation, however, can be predicted with a reasonable degree of accuracy by means of the empirical DuÈhring rule: the boiling point of a solution is a linear function of the boiling point of the pure solvent. Therefore, if the boiling points of solutions of different concentrations are plotted against those of the solvent at different pressures, a family of straight lines (not necessarily parallel) will be obtained. A typical DuÈhring plot, for aqueous solutions of sodium hydroxide, is given in Figure 2.9 from which it can be estimated, for example, that at a pressure at which water boils at 80  C, a solution containing 50 per cent by mass of NaOH would boil at about 120  C, i.e. a boiling point elevation of about 40  C. The boiling point elevation (BPE) of some 40 saturated aqueous solutions of inorganic salts have been reported by Meranda and Furter (1977) who proposed the correlating relationship

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Figure 2.9. DuÈhring plot for aqueous solutions of sodium hydroxide

BPE ( C) ˆ 104:9xS1:14

(2:41)

where xS is the mole fraction of salt in the solution. The practical difficulties of measuring boiling point elevations have been discussed by Nicol (1969). 2.11.1 Melting points The melting point of a solid organic substance is frequently adopted as a criterion of purity, but before any reliance can be placed on the test, it is necessary for the experimental procedure to be standardized. Several types of melting point apparatus are available commercially, but the most widely used method consists of heating a powdered sample of the material in a glass capillary tube located close to the bulb of a thermometer in an agitated bath of liquid. The best type of glass tube is about 1 mm internal diameter, about 70 mm long, with walls about 0.1 mm thick. The tube is heat-sealed at one end, and the powdered sample is scraped into the tube and knocked or vibrated down to the closed end to give a compacted layer about 5 mm deep. It is then inserted into the liquid bath at about 10  C below the expected melting point and attached close to the thermometer bulb. The bath is agitated and temperature is raised steadily at about 3  C/min. The melting point of the substance is taken as the temperature at which a definite meniscus is formed in the tube. For pure substances the melting point can be readily and accurately reproduced; for impure substances it is better to record a melting range of temperature.

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Table 2.8. Melting points of pure organic compounds useful for calibrating thermometers Substance Phenyl salicylate (salol) p-Dichlorbenzene Naphthalene m-Dinitrobenzene Acetamide Benzoic acid Urea Salicylic acid Succinic acid Anthracene p-Nitrobenzene acid Anthraquinone

Melting point,  C 42 53 81 90 114 123 133 160 183 217 242 286

Some organic substances are extremely sensitive to the presence of traces of alkali in soft soda-glass capillary tubes, giving unduly low melting points. In these cases, borosilicate glass tubes are recommended. Traces of moisture in the tube will also lower the melting point so the capillary tubes should always be stored in a desiccator. Some organic substances begin to decompose near their melting point, so that it is important not to keep the sample at an elevated temperature for prolonged periods. The insertion of the capillary into the bath at 10  C below the melting point, allowing the temperature to rise at 3  C/min, usually eliminates any difficulties. For reproducible results to be obtained, the sample should be in a finely divided state (<100 mm). The thermometer, preferably graduated in increments of 0.1  C, should be accurately calibrated over its whole range and, if in constant use, should be checked regularly. The wellknown standardization temperature are the freezing and boiling points of water, but other standards that can be used are the melting points of pure organic substances, such as those indicated in Table 2.8. The liquid used in the heating bath depends on the working temperature; water is quite suitable for melting points from about room temperature to about 70  C and liquid paraffin and a range of silicone fluids are widely used for more elevated temperatures. Specifications of methods for determining melting points are given in the British Pharmacopoeia (2000). See also section 4.5.1.

2.12 Enthalpies of phase change When a substance undergoes a phase change, a quantity of heat is transferred between the substance and its surrounding medium. Several types of enthalpy

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change, still often referred to as the latent heat, and the following types may be distinguished: solid I solid solid liquid

„ solid II „ liquid „ gas „ gas

(enthalpy (enthalpy (enthalpy (enthalpy

of of of of

transition,
Ht ) fusion,
Hf ) sublimation,
Hs ) vaporization,
Hv )

Only the last three of these represent significant quantities of heat energy: enthalpies of transition can for most industrial purposes be ignored. For example, the transformation of monoclinic to orthorhombic sulphur is accompanied by an enthalpy change of about 2 kJ kg 1 , whereas the fusion of orthorhombic sulphur is accompanied by an enthalpy change of about 70 kJ kg 1 . Enthalpy changes, like the other thermal properties, can be expressed on a mass or molar basis, but to avoid confusion, all in this section will be expressed on a molar basis. The relationship between some commonly used units are: 1 cal g

1

1 cal mol

ˆ 1 chu 1b 1

1

ˆ 1:8 Btu 1b

ˆ 1 chu 1b-mol

1

1

ˆ 4:187 kJ kg

ˆ 1:8 Btu 1b-mol

1

1

ˆ 4:187 J mol

1

The relationship between any enthalpy change
H and the pressure± volume±temperature conditions of a system is given by the Clapeyron equation dp
H ˆ dT T
v

(2:42)

where dp/dT ˆ rate of change of vapour pressure with absolute temperature and
v ˆ volume change accompanying the phase change. A typical temperature±pressure phase diagram for a one-component system is shown in Figure 2.10. The sublimation curve AX indicates the increase of the vapour pressure of the solid with an increase in temperature. This is expressed quantitatively by the Clapeyron equation written as

Figure 2.10. Temperature±pressure diagram for a single-component system

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 dp
Hs ˆ dT s T(vg vs )

(2:43)

where vg and vs ˆ the molar volumes of the vapour and solid, respectively, and
Hs ˆ enthalpy of sublimation. The vaporization curve XB indicates the increase of the vapour pressure of the liquid with an increase in temperature:   dp
Hv (2:44) ˆ dT v T(vg v1 ) where v1 ˆ the molar volume of the liquid, and
Hv ˆ latent heat of vaporization. Curve XB is not a continuation of curve AX; this fact can be confirmed by calculating their slopes dp/dT from equations 2.43 and 2.44 at point X. The fusion line XC indicates the effect of pressure on the melting point of the solid; it can either increase or decrease with an increase in pressure, but the effect is so small that line XC deviates only slightly from the vertical. When the fusion line deviates to the right, as in Figure 2.10, the melting point increases with an increase in pressure, and the substance contracts on freezing. Most substances behave in this manner. When the line deviates to the left, the melting point decreases with increasing pressure, and the substance expands on freezing. Water (see section 4.4) and type metals (Pb±Bi alloys) are among the few examples of this behaviour that can be quoted. The equation for the fusion line is   dp
Hf (2:45) ˆ dT t T(v1 vs ) where
Hf is the enthalpy of fusion. The enthalpies of sublimation, vaporization and fusion are related by
Hs ˆ
Hf ‡
Hv

(2:46)

but this additivity is applicable only at one specific temperature. The variation of an enthalpy change with temperature can be calculated from the Clausius equation d
H dT


H ˆ c2 T

c1

(2:47)

When
H ˆ
Hv , c1 and c2 are the molar heat capacities of the liquid, just on the point of vaporization, and of the saturated vapour, respectively. Equation 2.47 can also be used for calculating
Hf and
Hs the appropriate values of c being inserted. The specific volumes v1 and vs are much smaller than vg ; equations 2.43 and 2.44 can therefore be simplified to dp
H ˆ dT Tvg

(2:48)

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where
H ˆ
Hv or
Hs . From the ideal gas laws vg ˆ RT/p, so that equation 2.48 may be written dp
H ˆ dT p RT 2

(2:49)

This is the Clausius±Clapeyron equation. If the enthalpy change is considered to be constant over a small temperature range, T1 to T2 , equation 2.49 may be integrated to give ln

p2
H(T2 T1 ) ˆ RT1 T2 p1

(2:50)

Equation 2.50 can be used to estimate enthalpies of vaporization and sublimation if vapour pressure data are available, or to estimate vapour pressures from a value of the enthalpy change. Analysis of sublimation problems (see section 7.4), is frequently difficult owing to the scarcity of published vapour pressure and enthalpy data. If two values of vapour pressure are available, however, a considerable amount of information can be derived from equation 2.50 as illustrated by the following example. Suppose that the only data available on solid anthracene are that its vapour pressure at 210 and 145  C are 40 and 1.3 mbar, respectively; the vapour pressure at 100  C is required. Equation 2.50 can be used twice ± first to calculate a value of
Hs then that of the required pressure:   40
Hs (483 418) ln ˆ 1:3 8:314  483  418
Hs ˆ 88:5 kJ mol

1

Substituting this value of
Hs in equation 2.50 again,   40 88:5 (483 373) ln ˆ p100  C 8:314  483  373 therefore, p100  C ˆ 0:04 mbar 2.12.1 Enthalpy of vaporization There are several methods available for the estimation of enthalpies of vaporization at the atmospheric boiling point of the liquid. Trouton's rule, for example, is only suitable for non-polar liquids, but the Giacalone equation is fairly reliable for both polar and non-polar liquids:
Hvb ˆ 88Tb   RTc Tb
Hvb ˆ ln Pc Tc Tb

(Trouton)

(2:51)

(Giacalone)

(2:52)

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where
Hvb ˆ enthalpy of vaporization (J mol 1 ) at the boiling point, Tb ˆ boiling point (K) of the liquid at 760 mm Hg, Tc ˆ critical temperature (K) of the liquid, and Pc ˆ critical pressure (atm) of the liquid. (1 atm ˆ 760 mm Hg ˆ 1:013  105 N/m2 .) The enthalpy of vaporization of a liquid at some temperature T1 can be calculated from its value at another temperature T2 by means of the Watson equation:
Hv1

 Tc ˆ
Hv2 Tc

T1 T2

0:38

(2:53)

For example, the enthalpy of vaporization of benzene at its boiling point (353 K) is 30.8 J mol, its critical temperature 563 K. From equation 2.44 the corresponding value at 25  C (298 K) can be calculated as 33:6 J mol 1 , which compares with an experimental result of 33:7 J mol 1 . A critical account of these and other more recent methods is given by Reid, Prauznitz and Poling (1987).

2.13 Heats of solution and crystallization When a solute dissolves in a solvent without reaction, heat is usually absorbed from the surrounding medium (still commonly referred to as the heat of solution), i.e. if the dissolution occurs adiabatically the solution temperature falls. When a solute crystallizes out of its solution, heat is usually liberated (still commonly referred to as the heat of crystallization) and the solution temperature rises. The reverse cases, viz. heat evolution on dissolution and heat absorption on crystallization, may be encountered with solutes that exhibit an inverted solubility characteristic, e.g. anhydrous sodium sulphate in water. The dissolution of an anhydrous salt in water at a temperature at which the hydrated salt is the stable crystalline form frequently leads to the release of heat energy, owing to the exothermic nature of the hydration process: AB ‡ nH2 O ! AB
nH2 O Table 2.9 lists the heats of solution of anhydrous and hydrated magnesium sulphate and sodium carbonate in water to illustrate the effect of water of crystallization. The enthalpy changes associated with dissolution (
Hsol ) and crystallization (
Hcrys ) are generally recorded as the number of heat units liberated by the system when the process takes place isothermally. According to this system of nomenclature, if an adiabatic operation is considered, the expression
Hsol ˆ ‡q (heat units per unit mass of solute) means that the solution temperature will increase;
Hsol ˆ q means that it will fall. The magnitude of the heat effect accompanying the dissolution of solute in a given solvent or undersaturated solution depends on the quantities of solute and solvent involved, the initial and final concentrations and the temperature at

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Table 2.9. Heats of solution of anhydrous and hydrated salts in water at 18  C and infinite dilution Salt Magnesium sulphate

Sodium carbonate

Formula

Heat of solution, kJ mol 1

MgSO4 MgSO4
H2 O MgSO4
2H2 O MgSO4
4H2 O MgSO4
6H2 O MgSO4
7H2 O Na2 CO3 Na2 CO3
H2 O Na2 CO3
7H2 O Na2 CO3
10H2 O

‡88.3 ‡58.6 ‡49.0 ‡20.5 ‡2.3 13.3 ‡23.3 ‡9.2 42.6 67.9

which the dissolution occurs. The standard reference temperature is nowadays generally taken as 25  C. The first differential heat of solution (heat of solution at infinite dilution), 1
Hsol , may be regarded as the heat liberated or absorbed when 1 mole of solute dissolves in a large amount of pure solvent. This is the value most generally recorded in the data handbooks. For inorganic salts in water, it normally lies between about 5 and 15 kcal mol 1 , i.e. about 20 to 60 kJ mol 1 . For organic substances in organic solvents, it normally lies between 1 and 5 kcal mol 1 d (about 5 to 20 kJ mol 1 ). The last differential heat of solution
Hsol is the amount of heat liberated or absorbed, when 1 mole of the solute dissolves in a large amount of virtually saturated solution. This is numerically equal to the heat of crystallization,
Hcrys , but of opposite sign. The relationship between these quantities is 1 d ‡
Hdil ˆ
Hsol
Hcrys ˆ
Hsol

(2:54)

In crystallization practice, however, it is usual to take the heat of crystallization as being equal in magnitude, but opposite in sign, to the heat of solution at infinite dilution, since this is the quantity most commonly available in the handbooks, i.e.
Hcrys '

1
Hsol

(2:55)

Few values of heats of dilution are available in the literature, especially for the higher concentration ranges usually associated with industrial practice, but this quantity is usually only a small fraction of the heat of solution. Furthermore, as the dilution of most aqueous salt solutions is exothermic, i.e. the concentration is endothermic, the true value of the heat of crystallization will be slightly less than that obtained by taking the negative value of the heat of solution alone. Therefore the calculated quantity of heat to be removed from a crystallizing solution will be slightly greater than the true value, and this small error can serve as a factor of safety in the design of cooling heat transfer equipment.

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An alternative way to evaluate the heat of solution, and hence to estimate the heat of crystallization, is to consider the effect of temperature on the solubility: the greater the effect the higher is the heat of solution of the solute as quantified by the relationship d d ln c
Hsol ' dT RT 2

(2:56)

d is the final differential heat of solution. where
Hsol

2.14 Size classification of crystals The most widely employed physical test applied to a crystalline product is the one by means of which an estimate may be made of the particle size distribution. Product specifications invariably incorporate a clause that defines, often quite stringently, the degree of fineness or coarseness of the material. For many industrial purposes the demand is for a narrow range of particle size; regularity results in the crystals having good storage and transportation properties, a freeflowing nature and, above all, a pleasant appearance. Terms such as `fine' and `coarse' are frequently used, although usually without definition, to describe crystalline and powdered materials. For pharmaceutical products, however, some guidance is available from recommendations in the British Pharmacopoeia (2000), based on sieve gradings. Coarse Moderately coarse Moderately fine Fine Very fine Microfine

all passes 1700 mm all passes 710 mm all passes 355 mm all passes 180 mm all passes 125 mm ,90% passes 45 mm

.40% .40% .40% .40% .40%

passes passes passes passes passes

355 mm 250 mm 180 mm 125 mm 45 mm

Some of the more important procedures associated with the characterization of particulate solids are outlined below. 2.14.1 Sampling The physical and chemical characteristics of a bulk quantity of crystalline material are determined by means of tests on small samples. These test samples must be truly representative of the bulk quantity; otherwise any results obtained will be grossly misleading or completely useless. Inefficient sampling followed by careful analysis in the laboratory constitutes a waste of everyone's time and effort. Sampling, which is a highly specialized skill, should be carried out by conscientious, well-trained personnel who are fully aware of the tests that are to be made on the sample, without having any direct interest in the outcome of the analyses. The actual technique employed for sampling will depend on the nature of the bulk quantity of material, its location, the properties to be tested, the accuracy

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required in the test, and so on. Difficulties may be encountered in the sampling of solids in containers after transportation, owing to the partial segregation of fine and coarse particles; the fines tend to migrate towards the bottom of the container, and thorough remixing may be the only answer. Similar problems caused by segregation may be met in the sampling of solids flowing down chutes or through outlets. Hand sampling, widely employed for batch-produced or stored materials, is time-consuming and prone to error, but its use cannot always be avoided. One common method involves the use of a sample `thief ', a piece of pipe with a sharp bottom edge which is plunged into the full depth of the material. It is then withdrawn and the sample removed. This operation can be performed at fixed or random intervals in the bulk quantity. Sampling at intervals by means of scoops or shovels, known as grab sampling, is also widely used, but serious errors can be encountered when dealing with non-homogeneous materials. Automatic sampling is preferred to sampling by hand, and is also better suited to continuous processes. Ideally, the sample should be taken from a moving stream of solids or slurry. To eliminate segregation effects, samples should be taken by collecting the whole of the flowing stream for short periods. Isokinetic sampling is recommended for crystal suspensions in agitated vessels and pipelines (section 9.2). Bulk samples, which may range up to several hundred kilograms for large tonnage lots, have to be reduced to a smaller laboratory sample which, in turn, will have to be divided into several smaller test samples for subsequent analysis. These operations may be carried out by hand or with the aid of a sample divider. The best-known hand method is that of coning and quartering. The sequence of operations, carried out on a clean, smooth surface, or on glossy paper for small quantities in the laboratory, is shown in Figure 2.11. The bulk sample is thoroughly mixed and piled into a conical heap. The pile is then flattened and the truncated cone divided into four equal quarters (Figure 2.11c). This may be done, for example, with a sharp-edged wooden or sheet metal cross pressed into the heap. One pair of opposite quarters are rejected, the other pair are thoroughly mixed together and piled into a conical heap, the procedure being repeated until the required laboratory sample is obtained. A simple sample divider is the riffle which usually takes the form of a box divided into a number of compartments with bottoms sloping about 60 to the horizontal, the slopes of alternate chutes being directed towards opposite sides of the box. Thus, when the bulk sample is poured through the riffle, it is divided

Figure 2.11. Method of coning and quartering: (a) bulk sample in a conical heap; (b) flattened heap; (c) flattened heap quartered; (d ) two opposite quarters mixed together and piled into a conical heap

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Figure 2.12. Battery of riffles used for reduction of a large quantity of material

into two equal portions. The dimensions of a riffle depend on the size of the particles and the quantity of material ± an even flow of solids must be spread over the whole inlet area. When very large bulk samples have to be reduced to small test quantities, a battery of riffles decreasing in size may be employed, as shown diagrammatically in Figure 2.12. This arrangement is also suitable for the continuous or intermittent sampling of materials flowing out of hoppers or other items of process plant. The rotary sample divider, or spinning riffle, is less prone to operator error than is the static riffle. Basically it consists of a hopper which allows particulate material to flow on to a vibrating chute which then discharges into a number of sample boxes located in a rotating ring. Several units of this type are available commercially. The statistical theories of sampling are discussed by Allen (1990) who also describes, in considerable detail, a large number of sampling methods. BS 3406/1 (1986) also gives guidance on sampling, the sub-division of laboratory samples and the reporting of results. 2.14.2 Particle size and surface area A large number of methods are now available for measuring particle size, some of which are listed in Table 2.10 together with their approximate ranges of application. The book by Allen (1990) and a series of British Standards (BS 3406, 1986; BS 4359, 1984) give excellent coverage of the subject. Only a short review will be given here. It is not possible to measure or define absolutely the size of an irregular particle, and perfectly regular crystalline solids are rarely, if ever, encountered. The terms length, breadth, thickness or diameter applied to irregular particles are meaningless unless accompanied by further definition, because so many different values of these quantities can be measured. The only meaningful properties that can be defined for a single solid particle are the volume and surface area, but even the

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Table 2.10. Some methods of particle size measurement and their aproximate useful size ranges Method

Size range ( mm)

Sieving (woven wire) 125 000±20 Sieving (electroformed) 120±5 Sieving (perforated plate) 125 000±1000 Microscopy (optical) 150±0.5 Microscopy (electron) 5±0.001 Sedimentation (gravity) 50±1 Sedimentation (centrifugal) 5±0.1 Electrical sensing zone (Coulter) 200±1 Laser light scattering (Fraunhofer) 1000±0.1  Permeametry Surface area measurement: useful for particle sizes smaller than Gas adsorption about 50 mm

measurement of these quantities may present insuperable experimental difficulties. All particle size measurements are made by indirect methods: some property of the solid body which can be related to size is measured. Despite these difficulties of definition and measurement it is most convenient for classification purposes, if a single-length parameter can be ascribed to an irregular solid particle. The most frequent expression used in connection with particle size is the `equivalent diameter', i.e. the diameter of a sphere that behaves exactly like the given particle when submitted to the same experimental procedure. Several of these equivalent diameters are defined below. Sieving Woven wire test sieves were formerly designated by a mesh number (the number of wires per inch) but as the important sieve characteristic is the size of its apertures all standard test sieves are now designed, by international agreement, by their aperture size in millimetres or micrometres. The aperture sizes in a standard series are related to one another, e.g. following a fourth root of two (1.189) or a tenth root of ten (1.259) progression. The two most widely used standard sieve scales are the American (ASTM E11, 1995) and British (BS 410, 2000) both of which are compatible with the international scale (ISO 3310, 2000) (Table 2.11). The range of aperture sizes in most standard series extends from 125 mm to 20 mm. At the top end of the range particles must be carefully hand-placed on the sieve. At the lower end, sieving with the aid of a liquid is often needed to assist the flow of particles through the mesh. Particles that pass through a sieve are characterized by an equivalent sieve aperture diameter, ds:a: , the diameter of a sphere that would just pass through. Care needs to be taken to interpret this quantity, however, as explained in section 2.14.3 (see Figure 2.14). Perforated plate sieves are available, with round (125 to 1 mm) or square (125 to 4 mm) apertures, for coarse particle sizing. Microsieves with electroformed

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Table 2.11. Comparison between the British and US standard wire mesh sieve scales* showing the danger of using the `mesh number' as a sieve designation Mesh numbery (obsolete) 3 3 12 4 5 6 7 8 10 12 14 16 18 20 22 25 30 35 36 40 44 45

Aperture width (mm) BS 410 5600 4750 4000 3350 2800 2360 2000 1700 1400 1180 1000 850 710 600 500 425 355

ASTM E11

5600 4750 4000 3350 2800 2360 2000 1700 1400 1180 1000 850 710 600 500 425 355

Aperture width (mm) BS ASTM Mesh 410 E11 numbery (obsolete) 50 52 60 70 72 80 85 100 120 140 150 170 200 230 240 270 300 325 350 400 450

300 250 212 180 150 125 106 90 75 63 53 45 38 32

300 250 212 180 150 125 106 90 75 63 53 45 38 32

*Both standard test sieve scales BS 410 and ASTM E11 are compatible with the international (ISO 3310) scale. y The definition of `mesh number' is the number of apertures per inch in the sieve mesh. This obsolete designation leads to confusion because different standards specify different wire diameters.

round or square apertures (120 to 5 mm) in nickel plate are available for very fine particle sizing. Sieving is basically a very simple and justifiably popular particle sizing technique, but the precautions necessary to produce reliable data do not appear to be widely appreciated. Some of the more important points to note about the use of standard test sieves for particle size analysis are as follows. 1. Particles must not be forced through the sieve apertures. 2. Sieving should be continued to an end-point, i.e. until the amount of material passing through ceases to affect the result significantly. When using a mechanical shaker, it is recommended that each sieve removed from the stack should be given a brief brisk tapping and shaking by hand to ensure that the end-point has been reached. If it has not, sieving must be continued. 3. It should be clearly understood that the aperture size marked on the sieve is only a nominal size. The actual value can vary from this value, within

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specified tolerances. For accurate work, it is advisable to calibrate the sieve, e.g. by sieving a reference sample of known size distribution. Procedures for test sieving, both wet and dry, are prescribed in standard specifications (ISO 2591, 1989; BS 1796, 1990). Methods for the analysis of sieve test data are described in section 2.14.4. Microscopy Microscopy is commonly used as a basic reference method for particle sizing since individual particles may be observed while measuring or assessing their size, shape and composition. Particle images may be viewed directly in an optical microscope or by projection. The particle size may be recorded as the projected area diameter dp:a: , the diameter of a circle that has the same area as the projected image of the particle viewed in a direction perpendicular to its plane of maximum stability. This may be assessed by comparison with graduated circles on an eyepiece graticule. The microscopic method can be tedious and time-consuming, although automatic counting devices are now available. Photographic methods are popular, but can introduce further errors into the system (BS 3406/4, 1990; ISO 13322, 2001). The problems of preparing a microscope slide containing a well-dispersed representative sample of small crystals can be very considerable (Allen, 1990). Sedimentation A simple sedimentation technique, which readily lends itself to the determination of crystal size distribution in the range 1±50 mm, is the Andreasen pipette method. Although it is generally better to prepare a fresh suspension of the crystals under test in a suitable inert liquid, it is possible to classify crystals suspended in their own mother liquor. If the difference in density between the particles and suspending liquid is < 0:5 g cm 3 special care must be taken to avoid convection currents. The method, briefly, is as follows (BS 3406/2, 1986). A homogeneous suspension of the crystalline material in a suitable liquid is prepared in the graduated sedimentation cylinder of capacity 600 cm3 (Figure 2.13). Small samples (e.g. 10 cm3 ) of the suspension are withdrawn through the fixed pipette, at a known depth, h, below the liquid level, at chosen time intervals. The samples, including the one taken at zero time, are analysed for total suspended solids content by a suitable method. Ideally the suspension should be dilute (< 3 per cent) and a dispersion agent may be needed to prevent agglomeration: for particles in insoluble water a 0.1 per cent solution of sodium pyrophosphate is generally suitable. A sample taken at time t will contain no particles larger than size dSt calculated from Stokes' law which may be written 

18h dSt ˆ (s f )gt

1/2

(2:57)

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Figure 2.13. Andreasen fixed pipette method

Thus by taking samples at suitable intervals, e.g. 0, 5, 10, 20, 40, 80 . . . min, the size distribution of the original suspension may be evaluated. For routine analysis only one or two samples may be needed to characterize the particles. If t is measured in minutes, h in cm,  in g cm 3 and  in centipoise, then the particle size d, in mm, is given by  1/2 h dSt ˆ 17:5 (2:58) (s f )t For a given sample, n, the cumulative mass percentage, Pn , of particles smaller than the limiting Stokes' diameter for the time interval, tn , may be calculated from the mass Wn , of the suspended solids in the fraction by   Wn V
Pn ˆ 100 (2:59) W Vn where W ˆ mass (g) of solids originally suspended in the apparatus, V ˆ original volume of the suspension (cm3 ) and Vn ˆ volume of sample taken via pipette (cm3 ). A typical analysis is given in Table 2.12, where the size distribution of precipitated calcium carbonate (s ˆ 2:7 g cm 3 ) is measured by sedimentation at 20  C in water containing 0.1 per cent sodium pyrophosphate as dispersant (f ˆ 1:0 g cm 3 and  ˆ 1:0 cP, i.e. 10 3 N s/m2 ). In this test the CaCO3 was determined volumetrically by adding 0.2 M HCl to each sample, boiling to remove CO2 and back-titrating with 0.1 M NaOH. A `blank' was run on the suspending liquid. Alternatively, in this case, a gravimetric method could have been used, i.e. by evaporation to dryness. Descriptions of other gravitational sedimentation techniques are outlined in a recent international standard (BS ISO 13317, 2000) for particles in the size

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Table 2.12. Measurement of the particle size distribution of a sample of precipitated calcium carbonate by the Andreasen pipette method Time, t (min)

Pipette depth, h (cm)

Stokes' diameter, d (mm)

CaCO3 in fraction Wn (g)

Cumulative percentage undersize

0 5 10 20 40 80 160 320

20.0 19.6 19.2 18.7 18.3 17.9 17.4 17.0

± 28 19 13 9.1 6.4 4.4 3.1

0.231y 0.147 0.108 0.0763 0.0508 0.0299 0.0184 0.00924

± 64 47 33 22 13 8 4

*Sample volume Vn ˆ 10 cm3 . y Test sample mass W ˆ 14:3 g in suspension volume V ˆ 620 cm3 .

range 1±100 mm. For particles smaller than about 5 mm, however, problems can arise from convection effects and Brownian motion, but these difficulties may be reduced by speeding up the settling process by centrifuging the suspension. A number of procedural methods and commercial equipment for centrifugal sedimentation are now available for determining particle size in the 0:1 5 mm range (Allen, 1990; ISO 13318, 2000). Electrical sensing zone (Coulter) methods In the Coulter technique, particles have to be suspended in an electrolyte solution and then induced to pass through a small orifice, with surrounding electrodes, located in the measurement cell. Changes in electrical impedance in the orifice channel for each particle passage are measured and counted. The result is a number±size (volume) distribution of particles (BS ISO 13319, 2000). The method has found applicability in a wide range of industries. For application to crystallizing systems, however, it is important to choose a unit in which the voltage between the inner and the outer electrode is automatically adjusted so as to maintain a constant current. This renders the calibration, and hence the actual counting, insensitive to the type of electrolyte used as well as to concentration and temperature changes within the electrolyte. This precaution is of paramount importance for crystallization studies where changes in electrolyte properties due to phase transitions are inevitable (JancÏic and Grootscholten, 1984). Laser light scattering (Fraunhofer) methods Another widely used particle size analyser is based on the forward scattering of laser light through a dilute (<1% by volume) suspension of crystals retained in a small (10 mL) agitated cell. The resulting Fraunhofer diffraction pattern is detected and translated, by means of the instrument software, into a particle size distribution (BS ISO 13320, 2000).

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Specific surface area measurement In many cases, particularly for very small particles, surface area is a more appropriate characteristic to assess than some size based on an equivalent diameter. Particle surface area is important, for example, in paints and pigments or when chemical reactivity is an important property, as in the setting of cement. Precipitated materials are often characterized in this manner. Amongst the several techniques available, those based on permeability and gas adsorption are probably the most popular. In the permeability methods a known quantity of air is forced through a small bed of the fine solids under a constant pressure drop, and the flow time is recorded. The theory is based on the laminar flow of fluids through porous beds, and the specific surface area S (m2 g 1 ) of the material is calculated from the Kozeny equation S2 ˆ


P "3
ku L2 (1 ")2

(2:60)

where
P ˆ pressure drop across the bed; " ˆ voidage of the bed; L ˆ depth of the bed;  ˆ viscosity of the air; u ˆ empty-tube velocity;  ˆ density of the solid material; k is a constant (Kozeny's constant), which has a value equal to about 5.0 for granular solids. Several different types of permeability cell are available (BS 4359/2, 1982; Allen, 1990). A solid particle exposed to a gas will adsorb gas molecules on to its exposed surfaces. The derivation of a multilayer adsorption theory for gases on solid surfaces by Brunauer, Emmett and Teller in 1938 led to the development of the so-called BET adsorption methods for measuring the specific surface area of particulate solids. Several techniques are available (BS 4359/1, 1982; Lowell and Shields, 1984; Allen, 1990). Dry tests like the BET adsorption method can often give misleading information when used to characterize precipitated materials because the sample preparation operations of filtering, washing and drying can result in considerable damage and distortion to the particles. For this reason, a dye adsorption technique has been found to give a more realistic measurement of the specific area of precipitates while still suspended in their original mother liquor (Mullin et al., 1989a). A sample of the precipitate suspension is pipetted into a small flask containing a known quantity of a concentrated solution of a suitable dye, the selection of which depends on the chemical nature of the precipitate. After shaking to allow the mixture to come to equilibrium it is clarified in a laboratory centrifuge and the clear liquid analysed with a UV spectrophotometer. Knowing the mass of dye adsorbed on a given mass of precipitate, and the `coverage' value of the dye (the so-called Paneth value), it is a straightforward matter to calculate a specific surface area of the precipitate in, for example, m2 g 1 . It should also be noted that size data produced by the many different electronic techniques and instruments now available are dependent on the particular analysis algorithm incorporated into the instrument by the manufacturer. Significant differences in sizing results can therefore be recorded by

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commercial equipment of basically the same type. Indeed, for precise work, repeatability tests should be carried out to determine the closeness of agreement between independent tests made with the same method on identical test material in the same laboratory by the same operator using the same equipment. For inter-laboratory investigations reproducibility tests should be made to determine the closeness of agreement between tests made with the same method on identical test material in different laboratories with different operators using different equipment. Recommended procedures for repeatability and reproducibility tests are described in ISO 5725 (1992). Comparison of data From the above brief descriptions of a few commonly used particle sizing techniques it can be seen that the different methods arrive at an expression of particle `size' after measuring quite different particle properties. It is not surprising, therefore, that for a given sample of particulate material the different techniques will give different, often very different, sizes. It is advisable to use, if at all possible, only one sizing technique over the whole particle size range encountered in any one analysis. Even in sieving, only one type of sieving medium should be used throughout (woven wire, perforated or electroformed plate, round or square aperture) because of their individual sieving characteristics. When it is necessary, because of a very wide size distribution, to apply two different sizing techniques it is advisable that the data from both methods overlap over a significant part of the range to enable all the data to be converted to a common basis. Examples of overlap studies have been reported for sieve/zone sensing (Mullin and Ang, 1974; JancÏic and Grootscholten, 1984) and for sieve/laser light scattering (BrecÏevic and Garside, 1981). 2.14.3 Shape factors A precise calculation of the volume or surface area of a solid body of regular geometric shape can only be made when its length, breadth and thickness are known. For particulate solids in general, these three dimensions can never be precisely measured. Therefore, before a brief account is given of some of the methods of calculation available, a word of warning is necessary. It must be fully appreciated that the precision of calculation is always far greater than that of measurement of the various quantities used in the mathematical expressions. An equation, especially a complex one, always has a look of absolute dependability, but in this particular connection it most certainly leads to a false sense of security. All calculated volume or surface area data must be used with caution. Most calculation methods are based on one dimension of the particle, usually the equivalent diameter. If this dimension is obtained from a sieve analysis, it will be the sieve aperture diameter, ds:a: ; but as crystals are never true spheres, this diameter will normally be the second largest dimension of the particle. Figure 2.14 demonstrates some particle shapes that would, in a sieve analysis, all yield the same value for ds:a: . One potential source of error is thus clearly seen.

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Figure 2.14. Various particle shapes that would all be classified under the same sieve-aperture diameter

For a single particle, the size of which is defined by some length parameter or equivalent diameter, d, and density,  the following relationships can be applied: v ˆ fv d 3

(2:61)

m ˆ fv d 3

(2:62)

volume mass surface area

s ˆ fs d

2

(2:63)

The constants fv and fs may be called volume and surface shape factors, respectively. In the expressions for crystal growth rate in section 6.2.1, fV and fS are given the symbols  and , respectively. These latter symbols are also used in Chapter 8. For spherical (diameter ˆ d ) and cubic (length of side ˆ d ) particles   ˆ fv ˆ (sphere) and 1 (cube) 6 ˆ fs ˆ  (sphere) and 6 (cube) The shape factors are readily calculated for other regular geometrical solids. For p an octahedron, forpexample, with d representing the length of an edge, v ˆ (2)d 3 /3 and s ˆ 2 (3)d 2 , therefore p  ˆ fv ˆ v/d 3 ˆ (2)/3 ˆ 0:471 p ˆ fs ˆ s/d 2 ˆ 2 3 ˆ 3:46 From equations 2.61±2.63 two basic ratios may be defined: s fs d 2 F ˆ ˆ v fv d 3 d

(2:64)

s fs d 2 F ˆ ˆ m fv d 3 d

(2:65)

surface : volume surface : mass

Equation 2.65 defines the useful quantity known as the specific surface area, i.e. the surface area per unit mass of solid. The constant F ( ˆ fs /fv ˆ /) may be called the overall, surface±volume or specific surface shape factor. For spheres p F ˆ 6. For other shapes F > 6. For an octahedron p and cubes, fv ˆ (2)/3, fs ˆ 2 3 and F ˆ 7:35. Values of F  10 are frequently encountered in comminuted solids, and much higher values may be found for flakes and plate-like crystals. If the particles are elongated or needle-shaped, their volume and surface area may be calculated on the assumption that they are

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cylindrical; length and diameter may be measured microscopically, or the diameter can be taken as the equivalent sieve aperture diameter. For example, a crystal with a length : breadth : thickness ratio of 5 : 2 : 1 would be characterized in sieving by its second largest dimension, i.e. d ˆ 2. Therefore fv ˆ v/d 3 ˆ 10/8 ˆ 1:25, fs ˆ s/d 2 ˆ 34/4 ˆ 8:5 and F ˆ fs /fv ˆ 6:8. Volume shape factors may be measured by weighing a known number of particles from a close-sieve fraction, the number depending on the size of the particles and the accuracy with which the total mass can be weighed. The need for a close-sieve fraction arises from the fact that the shape factor of crystals can vary greatly with size (Garside, Mullin and Das, 1973). The determination of volume shape factors for particles smaller than about 500 mm becomes extremely difficult, since it may be necessary to count and weigh several thousand particles. However, the following method may be used to simplify the procedure. Prepare a sample of the particles by sieving between two close sieves. Clean the finer of the two sieves (the retaining sieve) and attach a strip of adhesive tape of known mass and dimensions, to its underside. Place a quantity of the particles on the sieve and shake the sieve for several minutes. Peel off the adhesive tape, which will now have hundreds or thousands of particles in a regular matrix (more or less one per sieve aperture). The approximate number of particles per unit area can be determined from the designation of the sieve mesh. For example, a 150 mm aperture sieve with 100 mm diameter wires contains about 1600 apertures per cm2 . The adhesive strip can then be weighed and the average mass of one particle determined. Surface shape factors are much more difficult to measure than volume shape factors and they are subject to greater uncertainty. One method is as follows. A few individual crystals are observed through a low-power microscope fitted with a calibrated eyepiece, and sufficient measurements taken to allow a sketch to be made of a representative geometric shape, e.g. a parallelepipedon, ellipsoid, oblate spheroid, etc. The surface area of the representative solid body may then be calculated. It should be appreciated, of course, that the result of such a calculation will be prone to significant error. Another quantity that has been used to characterize crystal shape is the sphericity, , defined as the ratio of the surface area of a sphere having the same volume as the particle to the apparent estimated surface area of the particle. This can be rewritten (NyÂvlt, 1990) as ˆ

(6fV /)2/3 fS /

(2:66)

For isometric particles is close to 1 while for needles or platelets its value is much lower. Evaluation of is useful for checking the values of fV and fS since 0 < < 1. When three mutually perpendicular dimensions of a particle may be determined, Heywood's ratios may help to characterize shape (Allen, 1990): elongation ratio n ˆ L/B flakiness ratio m ˆ B/T

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The thickness T is the minimum distance between two parallel planes which are tangential to opposite surfaces of the particle, one plane being the plane of maximum stability. The breadth B is the minimum distance between two parallel planes which are perpendicular to the planes defining the thickness and are tangential to opposite sides of the particle. The length L is the distance between two parallel planes which are perpendicular to the planes defining thickness and breadth and are tangential to opposite sides of the particle. When using shape factors, it is important to remember that they depend on the dimension chosen to characterize the particle. For example, a geometrical shape such as regular parallelepipedon with a length : breadth : thickness ratio, L : B : T, of 5 : 2 : 1 the following shape factors may be calculated:

using L ˆ 5 using B ˆ 2 using T ˆ 1

fv

fs

F

0.08 1.25 10

1.4 8.5 34

17 6.8 3.4

Different answers again would result if measured values of the different equivalent diameters, such as ds:a: , dp:a: , dSt , etc. were used. An example of apparent shape factor change caused by the use of different sieves (woven wire, round or square hole, perforated plate) to measure the particle size is reported by Garside, Mullin and Das (1973). 2.14.4 Size data analysis Mean particle size In a total mass, M, of uniform particles, each of mass m and equivalent diameter d, the number of particles, n, is given by nˆ

M M ˆ m fv d 3

(2:67)

and the total surface area, s, by s ˆ ns ˆ

fs Md 2 FM ˆ d fv d 3

(2:68)

Equations 2.67 and 2.68 can be applied to masses of non-uniform particles if a suitable average or mean value of d can be chosen. This becomes an intractable problem, however, since there are so many possible choices that could be made. The simplest of all average diameters is the arithmetic mean. For example, if sieving has been carried out between two sieves of aperture a1 and a2 , the average particle equivalent diameter is given by da ˆ (a1 ‡ a2 )/2

p 4

(2:69)

This description may be quite adequate for two close sieves in a 2 series, but it can be absolutely meaningless for two sieves at extreme ends of the mesh range. Another simple average diameter is the geometric mean, defined by

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Physical and thermal properties

p dg ˆ (a1 a2 )

77

(2:70)

Values of d calculated from equation 2.70 are smaller than those given by equation 2.67, but for two close sieves the difference is not great. The volume mean diameter (or mass mean if the particle density is constant) is widely used: nd 4 (Md ) dv ˆ ˆ M nd 3

(2:71)

When the surface area of the particles is an important property the surface mean diameter can be employed, defined by nd 3 M ds ˆ ˆ nd 2 (M/d)

(2:72)

where n and M are the number and mass, respectively, of all particles of equivalent diameter d. The root mean square diameter is also frequently used when surface properties are important. This statistical quantity is defined by s  r  nd 2 (M/d )  drms ˆ (2:73) ˆ (M/d 3 ) n Values of the overall mean diameters calculated from equations 2.71±2.73 can differ considerably (see Table 2.13), yet for a mass of particles with a wide size distribution there is no general agreement as to the preferred method. Two other statistical diameters are often encountered, viz. the modal and median diameters; both are determined from frequency plots (size interval versus number of particles in each interval). The modal diameter is the diameter at the peak of the frequency curve, whereas the median diameter defines a midpoint in the distribution ± half the total number of particles are smaller than the median, half are larger. If the distribution curve obeys the Gaussian or Normal Error law, the median and modal diameters coincide. Table 2.13. Calculation of overall `mean' diameters Size range (mm) 850±600 600±425 425±300 300±212 212±150

M/d 3

Mean size of fraction, d (mm)

Mass of fraction, M (g)

Md

M/d

725 512 362 256 181

11.8 18.6 38.5 22.7 8.4

8550 9520 13 900 5810 1520

0.0163 0.0363 0.1064 0.0887 0.0464

0:031  10 0:139  10 0:812  10 1:353  10 1:417  10

6

100.0

39 300

0.2941

3:752  10

6

dv ˆ 39 300/100 ˆ 393 mm. ds ˆ 100/0:2941 ˆ 340 mm. drms ˆ (0:2941/3:75  10 6 )1/2 ˆ 280 mm.

6 6 6 6

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In connection with particle size measurement more than 20 different `mean' diameters have been proposed at one time or another; and while several have certain points in their favour in special cases, none has yet been found to be generally satisfactory. Therefore all calculations based on an average diameter are prone to appreciable error, and it is recommended that such calculated quantities be clearly annotated with the method of calculation so that the results of different workers can be compared. Graphical analysis Once a size analysis has been performed and the results recorded, there remains the task of assessing the size characteristics of the tested material and of extracting the maximum amount of information from the data. While a table may record all the measured quantities, this form of expression is not always the best one; the magnitudes of the various quantities may be readily visualized, but certain trends may be completely obscured in a mass of figures. The real significance of a sizing test can most readily be judged when the data are expressed graphically. From such a pictorial representation trends in the data are easily detected, and the prediction of the expected behaviour of the material on sieves other than those used in the test can often be made with a reasonable degree of accuracy. Many different forms of graphical expression may be employed, and the use and applicability of some of these methods are demonstrated below with reference to the results of a sieve test. The graphical procedures described, however, are applicable, with suitable nomenclature changes, to all methods of particle size analysis. The sieve test data in Table 2.14 are recorded in three different ways, viz. the percentage by mass of the fractions retained on each sieve, and the cumulative mass percentages of oversize and undersize material. Four types of graph paper are commonly used for plotting particle size distributions, depending on the sort of information that is required: (a) ordinary Table 2.14. Sieve test data used for the construction of Figures 2.15 to 2.18 Sieve aperture, mm 2360 1700 1180 850 600 425 300 212 150 106 ±

Fractional mass per cent retained

Cumulative mass per cent oversize

Cumulative mass per cent undersize

1.2 2.9 18.8 28.8 22.0 11.1 6.0 3.9 1.8 1.3 2.2

1.2 4.1 22.9 51.7 73.7 84.8 90.8 94.7 96.5 97.8 ±

98.8 95.9 77.1 48.3 26.3 15.2 9.2 5.3 3.5 2.2 ±

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squared or arithmetic, (b) log±linear or semi-log, where one of the axes is marked off on a log scale and the other on an arithmetic scale, (c) log±log where both axes are marked off on a logarithmic scale, and (d) arithmetic±probability, where one axis is marked off on a probability scale, the intervals being based on the probability integral. Log±probability and double-logarithmic (RRS) grids also find use in special cases. In Figure 2.15a the mass percentages of the fractions retained on each successive sieve used in the test are plotted against the widths of the sieve apertures (in microns). The lines joining the points have no significance; they merely complete the frequency polygon. The sharp peak in the distribution curve occurs at 850 mm. This point, however, represents the fraction that passes the 1180 mm sieve and is retained on the 850 mm, so it could be plotted at the `mean' size of 1015 mm. Alternatively, the results may be represented in the form of a frequency histogram (Figure 2.15b) depicting the size range of each collected fraction. From both diagrams the general picture of the overall spread of particle size can be seen quite clearly, but the simple arithmetic method of plotting suffers from the disadvantage of producing a congested picture in the regions of the fine mesh sieves. When the data are plotted on semi-log paper (Figure 2.16a), with the aperture widths recorded on the logarithmic scale, the points in the coarse sieve region are brought closer together, and those in the fine sieve region located further apart than in the corresponding simple arithmetic plot. In fact, the successive points are more or less equally spaced along the horizontal scale. The frequency histogram (Figure 2.16b) is composed of columns of approximately equal widths.

Figure 2.15. Sieve test data plotted on arithmetic graph paper ± percentage by weight of the fractions retained between two given sieves in the B.S. series: (a) the frequency polygon; (b) the frequency histogram

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Figure 2.16. Sieve test data plotted on semi-log graph paper ± percentage by weight of fractions retained between two sieves in the B.S. series: (a) the frequency polygon; (b) the frequency histogram

The semi-log graphs of the cumulative oversize and undersize percentages (Figure 2.17a) show that the curves are mirror images of each other. In practice only one need be plotted. The two curves cross over at the median size (50 per cent is larger than the median, and 50 per cent is smaller). In this case the median size is 870 mm. Interpolation is facilitated by the even spread of the plotted points. It can be estimated, for instance, that about 87 per cent of the original material would be retained on a 355 mm sieve or that about 7 per cent would pass through a 250 mm sieve. The cumulative percentages of oversize and undersize particles may also be plotted against aperture size on a log±log basis (Figure 2.17b). In this type of plot the cumulative undersize data tend to lie on a straight line over a wide range of particle size, about 100 to 1200 mm in this case. The undersize and oversize curves are clearly not mirror images, and oversize data are rarely correlated on this basis. The log-log method of plotting of undersize data is extremely useful because rough checks may be made on the size distribution by the use of only two, or possibly three, test sieves. Material of the type considered in Table 2.11 could be size-checked with 1000, 500 and 250 mm sieves, for example. Coefficient of variation (CV) Probability plots have often been suggested for particle size analysis, particularly in connection with the assessment of comminution processes.

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Figure 2.17. Sieve test data plotted on (a) semi-log graph paper, (b) on log±log graph paper±cumulative oversize and undersize percentages

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The following method, proposed by Powers (1948) for use in the sugar industry, has much to be said in its favour for the size specification of crystalline products. This method employs arithmetic-probability graph paper (one scale divided into equal intervals, the other marked off according to the probability integral) and provides a simple means of recording the crystal size distribution in terms of two numbers only ± the median size (MS), and a statistical quantity the coefficient of variation (CV) expressed as a percentage. The evaluation of MS/CV is demonstrated in Figure 2.18a with the sieve analysis data from Table 2.13. The cumulative undersizes (or oversizes if preferred) are plotted on the probability scale, the sieve aperture sizes on the arithmetic scale. If the data between about 10 and 90 per cent lie on a straight line, the MS/CV method can be applied. The data in Figure 2.18a comply with this requirement. Thus the median size is 870 mm. The coefficient of variation can be deduced as follows. The equation for the normal probability (Gaussian) curve may be expressed as   2 1 (d d) f (d) ˆ p exp (2:74) 22  (2) where d is an equivalent particle diameter, in this case based on sieve aperture size, and  is the standard deviation. If the area enclosed under the normal curve between sieve apertures d ˆ 0 to 1 is taken as unity, the area enclosed between d ˆ 0 and d ˆ d ‡ , where d is the median (50 per cent) size MS, is 0.8413. This value is obtained from tables of the normal probability function. Therefore the area enclosed between d ˆ d ‡  and d ˆ 1 is 1 0:8413 ˆ 0:1587. The value of , the standard deviation, can be obtained from the arithmetic probability diagram by reading the value of d at 84.13 per cent (84 per cent is accurate enough for this purpose) and subtracting the value of  Alternatively, the value of d at 15.87 (or 16) per cent can be subtracted from d.  i.e. d, d ˆ d

 ˆ d84%

d16%

These two values of  may not coincide, so a mean value can be taken as ˆ

d84%

d16% 2

The coefficient of variation, as a percentage, is given by CV ˆ

100 100 ˆ d50% d

(2:75)

CV ˆ

100(d84% d16% ) 2d50%

(2:76)

or

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Figure 2.18. Sieve test data plotted on (a) arithmetic/probability graph paper (cumulative undersize percentages) illustrating the use of the MS/CV method of analysis, (b) on Rosin± Rammler±Sperling (RRS) double logarithmic graph paper

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From Figure 2.18a CV ˆ

100(1270 440) ˆ 48 per cent 2(870)

and the size distribution can be specified, in terms of MS/CV, as 870/48. The size distribution does not have to be Gaussian for the MS/CV method to be applied. Many skew distributions also give the necessary linear relationship between about 10 and 90 per cent, although in these cases MS will not coincide with the modal diameter at the peak of the distribution curve. For skew distributions that are not approximately Gaussian over the 10±90 per cent region, plotting on log-probability paper (one scale logarithmic, the other marked off according to the logarithmic probability function) may give a better correlation. The equation of the logarithmic normal curve is   1 ( log d log d0 )2 p exp f ( log d ) ˆ (2:77) log 0 2 2 log2 0 where d0 is the geometric mean size and 0 is the geometric standard deviation. The log-normal distribution gives a curve skewed towards the larger sizes, and it frequently gives a good representation of particle size distributions from precipitation and comminution processes. Furthermore, the log-normal distribution is often used because it overcomes the objection to the normal (Gaussian) distribution function which implies the existence of particles of negative size. Another distribution function gaining popularity for characterizing crystal size distributions is the gamma function, expressed as f (d) ˆ d x exp (dx/y) (x ‡ 1) ( y/x)x‡1

(2:78)

The parameters x and y, which give measures of the `skewness' and `size' of the distribution respectively, can be related to the crystallization process. The median size MS (ˆ d) and standard deviation, , may be calculated from d ˆ y(x ‡ 1)/x

and

 ˆ y(x ‡ 1)1/2 /x

Therefore from equation 2.75 the coefficient of variation is given by CV ˆ 100(x ‡ 1)

1/2

(2:79)

Some skew distributions, particularly those of comminuted materials can be fitted by the Rosin±Rammler±Sperling (RRS) function. This relationship, based on one originally derived from probability considerations, may be written P ˆ 100 exp[ (d/d 0 )n ]

(2:80)

where P ˆ cumulative percentage oversize, d ˆ particle size, and d 0 is a statistical mean size corresponding to P ˆ 36:8% (100/e, where e ˆ 2:718, the base of natural logarithms). Equation 2.80 indicates a linear relationship between

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log log(100/P) and log d. The slope of the line, n, has been called a `uniformity factor', i.e. n ˆ tan , where  is the angle between the RRS line and the horizontal. As the size distribution narrows towards a mono-sized dispersion, n ! 1; as it broadens, n ! 0. Special double logarithmic graph paper suitable for this type of plotting is available and the data from Table 2.14 are shown on such a plot in Figure 2.18b. From this plot it is possible to determine the 16, 50 and 84% cumulative percentages needed to calculate the MS/CV, as described above. Alternatively, the distribution can be characterized by the uniformity factor, n. In the example shown the median size d50% ˆ 870 mm (the same as determined in Figure 2.18a), the statistical mean size d 0 ˆ 1000 mm, and the uniformity factor n ˆ 1:8.

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3

Solutions and solubility

3.1 Solutions and melts A solution (gaseous, liquid or solid) is a homogeneous mixture of two or more substances. The constituents of liquid solutions are frequently called solvents and solutes, but despite common usage there is no fundamental reason why any one particular component of a solution should be termed the solvent, and considerable confusion can arise from adhering to rigid definitions. For example, a salt such as potassium nitrate fuses in the presence of small amounts of water at a much lower temperature than the pure salt does. The use of the term `solvent' for water would hardly seem to be justified in this case, and although it may seem strange to refer to `a solution of water in potassium nitrate', this would be an equally acceptable description. Fusion is nothing more than an extreme case of liquefaction by solution, so it may be said that when a salt dissolves in water the salt, in fact, melts. Owing to the widespread and often indiscriminate use of the word `melt', it is difficult to give a precise definition of the term. Strictly speaking, a melt is a liquid close to its freezing point, but in its general application the term also includes homogeneous liquid mixtures of two or more substances that would individually solidify on cooling to ambient temperatures. Thus -naphthol heated above its melting point (96  C) would be regarded as a melt, and so would a liquid mixture of -naphthol and -naphthol (m.p. 122  C). On the other hand, a liquid mixture of -naphthol and methanol would normally be classified as a solution. However, no rigid definition is possible. The KNO3 ±H2 O system quoted above, and the many well-known cases of hydrated salts dissolving in their own water of crystallization at elevated temperatures, would in all probability be considered to be melts.

3.2 Solvent selection Water is almost exclusively used as the solvent for the industrial crystallization of inorganic substances from solution. This fact is quite understandable because, apart from the relative ease with which a very large number of chemical compounds dissolve in it, water is readily available, cheap and innocuous. For these reasons water is used whenever possible even for the industrial crystallization of organic compounds, although for a variety of reasons other solvents may have to be used in this particular field. The selection of the `best' solvent for a given crystallization operation is not always an easy matter. Many factors must be considered and some compromise must inevitably be made; several undesirable characteristics may have to be accepted to secure the aid of one important solvent property. There are several

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hundred organic liquids that are potentially capable of acting as crystallization solvents, but outside the laboratory the list can be shortened to a few dozen selected from the following groups: acetic acid and its esters, lower alcohols and ketones, ethers, chlorinated hydrocarbons, benzene homologues, and light petroleum fractions. In many cases, of course, the solvent may already have been selected by the prevailing process conditions. In others, the cost of solvent recovery may override other considerations. A mixture of two or more solvents will occasionally be found to possess the best properties for a particular crystallization purpose. Common binary solvent mixtures that have proved useful include alcohols with water, ketones, ethers, chlorinated hydrocarbons or benzene homologues, etc. and normal alkanes with chlorinated hydrocarbons or aromatic hydrocarbons. A second liquid is sometimes added to a solution to reduce the solubility of the solute, cause its precipitation/crystallization and maximize the yield of product. It is necessary, of course, for the two liquids (the original solvent and the added precipitant) to be completely miscible with one another in all proportions. The process is commonly encountered, for instance, in the crystallization of organic substances from water-miscible organic solvents by the controlled addition of water. The term `watering-out' is often used in this connection. This approach is also used to reduce the solubility of an inorganic salt in aqueous solution by the addition of a water-miscible organic solvent in salting-out precipitation processes (section 7.2.5). Some of the main points that should be considered when choosing a solvent for a crystallization process include the following. The solute to be crystallized should be readily soluble in the solvent. It should also be easily deposited from the solution in the desired crystalline form after cooling, evaporation, saltingout with an additive, etc. There are many exceptions to the frequently quoted rule that `like dissolves like', but this rough empiricism can serve as a useful guide. Solvents may be classified as being polar or non-polar; the former description is given to liquids which have high dielectric constants, e.g. water, acids, alcohols, and the latter refers to liquids of low dielectric constant, e.g. aromatic hydrocarbons. A non-polar solute (e.g. anthracene) is usually more soluble in a non-polar solvent (e.g. benzene) than in a polar solvent (e.g. water). However, close chemical similarity between solute and solvent should be avoided, because their mutual solubility will in all probability be high, and crystallization may prove difficult or uneconomical. It should be noted that the crystal habit can often be changed by changing the solvent (section 6.4). Based on the nature of their intermolecular bonding interactions solvents may be conveniently divided into three main classes: 1. polar protic, e.g. water, methanol, acetic acid; 2. dipolar aprotic, e.g. nitrobenzene, acetonitrile, furfural; 3. non-polar aprotic, e.g. hexane, benzene, ethyl ether. In polar protic solvents the solvent molecules interact by forming strong hydrogen bonds. In order to dissolve, the solute must break these bonds and replace them with bonds of similar strength. To have a reasonable solubility, therefore, the solute must be capable of forming hydrogen bonds, either

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because the solute itself is hydrogen bonded or because it is sufficiently basic to accept a donated hydrogen atom to form a hydrogen bond. If the solute is aprotic and not basic it cannot form strong bonds with the solvent molecules and therefore will have a very low solubility. In dipolar aprotic solvents, characterized by high dielectric constants, the solvent molecules interact by dipole±dipole interactions. If the solute is also dipolar and aprotic it can interact readily with the solvent molecules forming similar dipole±dipole interactions. If the solute is non-polar it cannot interact with the dipoles of the solvent molecules and so cannot dissolve. Protic solutes are found to be soluble in basic dipolar aprotic solvents because strong hydrogen bonds are formed, replacing the hydrogen bonds between the solute molecules in the solid state. If a dipolar aprotic solvent is not basic, however, a protic solute will have a low solubility because the strong hydrogen bonds in the solid phase can only be replaced by weaker dipole±dipole interactions between solvent and solute molecules. In non-polar aprotic solvents, characterized by low dielectric constants, molecules interact by weak van der Waals forces. Non-polar solutes are readily soluble as the van der Waals forces between solute molecules in the solid phase are replaced by similar interactions with solvent molecules. Dipolar and polar protic solutes are generally found to have very low solubilities in these solvents except in cases where non-polar complexes are formed. Solvent power The `power' of a solvent is usually expressed as the mass of solute that can be dissolved in a given mass of pure solvent at one specified temperature. Water, for example, is a more powerful solvent at 20  C for calcium chloride than n-propanol (75 and 16 g/100 g solvent, respectively). At the same temperature n-propanol is a more powerful solvent than water for benzoic acid (42.5 and 0.29 g/100 g solvent, respectively). The temperature coefficient of solubility is another important factor to be considered. For example, at 20  C water is a more powerful solvent for potassium sulphate (11 g/100 g water) than for potassium chlorate (7 g/100 g), but the converse is true at 80  C (K2 SO4 , 21 g/100 g; KClO3 , 39 g/100 g). Thus, on cooling the respective saturated solutions from 80 to 20  C, more than 80% of the dissolved KClO3 would be deposited compared with less than 50% of the K2 SO4 . Both the solvent power and the temperature coefficient of solubility must be considered when choosing a solvent for a cooling crystallization process; the former quantity influences the volume of the crystallizer, and the latter determines the crystal yield. It frequently happens, especially in aqueous organic systems, that a low solubility is combined with a high temperature coefficient of solubility. For example, the solubilities of salicylic acid in water at 20 and 80  C are 0.20 and 2.26 g/100 g, respectively. Therefore, on cooling from 80 to 20  C, most of the dissolved solute (91%) is deposited, and consequently the solute yield is high. However, on account of the low solubility, even at the higher temperature, an excessively large crystallizer would be required to

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give a reasonable production rate and consequently water could not be considered as a suitable solvent for salicylic acid crystallization. Potassium chromate is an example of a solute with a reasonably high solubility in water and a low temperature coefficient of solubility (61.7 and 72.1 g/ 100 g at 20 and 80  C, respectively). The low yield on cooling (about 14 per cent) makes it necessary to effect crystallization in some other manner, such as a combination of cooling and evaporation, thus increasing the cost of the operation. An algorithm for the prediction of an optimal solvent or solvent mixture for cooling crystallization has been proposed by Nass (1994). Solvent hazards A few words on the subjects of purity and hazards might not be entirely out of place at this point, because their consideration is inevitable when choosing a solvent for a crystallization process. No deleterious impurity, dissolved or suspended, should be introduced into a crystallizing system. The solvent, therefore, should be as clean and as pure as possible. No colouring matter should be permitted to affect the appearance of the final crystals. No residual odours should remain in the product after drying, a problem often encountered after crystallization from organic solvents with distinctive odours. If no previous experience has been obtained with a potential solvent, simple laboratory trials should be made, but due caution should be exercised in interpreting the results because laboratory filtration, washing and drying techniques generally prove to be much more efficient than the corresponding large-scale operations. The solvent should be stable under all foreseeable operating conditions: it should neither decompose nor oxidize, and it must not attack any of the materials of construction of the plant. When organic solvents are being used, care must be taken in choosing the correct gasket materials; most common types of rubber and many synthetic elastomers, for example, swell and disintegrate after prolonged contact with chlorinated hydrocarbons. The solute and solvent should not be capable of reacting together chemically, although solvate formation may be permitted under certain circumstances. Hydrated crystals are frequently desired as end-products, but should the anhydrous substance be required the necessary drying process may prove difficult and expensive. Methanol, ethanol, benzene and acetic acid are also known to form solvates with certain substances, and the loss of solvent on drying imposes an additional cost on the process. Highly viscous solvents are not usually conducive to efficient crystallization, filtration and washing operations. In general, therefore, solvents of low viscosity are preferred. If the solvent recovery process involves distillation, a reasonably volatile solvent is desirable. On the other hand, the loss of a solvent with a high vapour pressure from filters and other processing equipment can be considerable and may prove both costly and hazardous. Solvents with freezing points above about 5  C present wintertime storage and transportation difficulties. Benzene (f.p. 5  C) and acetic acid (f.p. 17  C) are good examples. Most organic solvents employed in cystallization processes are flammable, and their use necessitates stringent operating conditions. Two of the most

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important properties of a flammable solvent are the flash point and explosive limits; the former is the temperature at which the mixture of air and vapour above the liquid can be ignited by means of a spark, and the latter refers to the percentages by volume in air between which the vapour mixture will, if ignited, explode in a confined space. Diethyl ether is an example of a solvent with a very low flash-point ( 30  C) and wide range of explosive limits (1 to 50%). All organic solvents are toxic to a greater or lesser degree: the prolonged inhalation of almost any vapour will produce some harmful effect on a human being. Some solvents are acute poisons, some have a cumulative poisoning effect, and others produce narcosis or intoxication on inhalation, or dermatitis on contact with the skin. Information on these aspects and on the maximum vapour concentrations permitted in working areas can be obtained from specialized reference books. The handbooks by Sax (1992) and Bretherick (1999) deal comprehensively with solvent properties and hazards. Health risks in the use of common solvents are dealt with in RSC/CEC (1986, 1988).

3.3 Expression of solution composition The composition of a solution, or melt, may be expressed in many different ways, e.g. mass per unit mass of solvent, mass per unit mass of solution, mass per unit volume of solvent, and so on. The mass unit may refer to the dissolved species itself or to a solvated form, e.g. a hydrate. For the expression of crystallization kinetics there is some theoretical justification for recording compositions on a molar basis, e.g. as kmol m 3 (i.e. mol L 1 ), while mole fractions are most frequently used for thermodynamic calculations. Mass fractions are commonly used in the construction of phase diagrams, although the use of mole fractions is recommended for the representation of reciprocal salt pair systems (section 4.7.2). For the purpose of expressing a mass balance on an item of process plant, there is considerable merit in expressing solution composition as mass of unsolvated solute per unit mass of solvent, particularly when temperature changes are expected. This avoids the need for further calculation to account for density changes. In view of the frequent need to make interconversions of composition units it is recommended that, whenever solution concentration measurements are made, the density of the solution at the relevant temperature is also measured and recorded (see section 3.9). Many of the above methods of solubility expression can lead to the use of the potentially misleading term `percentage concentration'. For instance, an expression such as `a 10 per cent aqueous solution of sodium sulphate' could be taken to mean, without further definition, any one of the following: 10 g 10 g 10 g 10 g

of Na2 SO4 in 100 g of water of Na2 SO4 in 100 g of solution of Na2 SO4
10H2 O in 100 g of water of Na2 SO4
10H2 O in 100 g of solution

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If 10 g of anhydrous Na2 SO4 in 100 g of water were the intended description of the solution concentration, this would then be equivalent to 9.1 g of Na2 SO4 in 100 g of solution 20.6 g of Na2 SO4
10H2 O in 100 g of solution 26.0 g of Na2 SO4
10H2 O in 100 g of water which gives some measure of the magnitude of the possible misinterpretation. To make matters even worse, the term `percentage concentration' is often applied on a volume basis, e.g. 10 g of Na2 SO4 in 100 mL of water, of solution, and so on. Table 3.1 lists the interconversions between a number of the common expressions of solution composition. For convenience, the expressions are drafted in terms of aqueous systems, but the relationships are completely general if the terms `unsolvated substance', `solvate' and `solvent' are substituted for `anhydrous substance', `hydrate', and `water', respectively. Table 3.1. Conversion factors for solution concentration units Concentration

Equivalent expressions

C1

1

C2

C1 1 ‡ C1

C3

RC1 1 ‡ C1

C4

1

C2 C2

R

C3 C4 C3 R ‡ (R 1)C4 C3 R

RC2



C5 C5

C4 R(1 ‡ C4 )

C5 

C6 R

MA C7 

C4 1 ‡ C4

RC5 

C6 

MH C7 

RC1 RC2 C3 (R 1)C2 1 RC2 1 C3



RC5 RC5

C5

C1 1 ‡ C1

C2

C3 R

C4 R(1 ‡ C4 )

C6

RC1 1 ‡ C1

RC2

C3

C4 1 ‡ C4

RC5

C1 MA (1 ‡ C1 )

C2 MA

C3 MH

C4 MH (1 ‡ C4 )

C5 MA

C7 and C8

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 MA MH MW R 

ˆ kg of anhydrous substance/kg of water ˆ kg of anhydrous substance/kg of solution ˆ kg of hydrate/kg of solution ˆ kg of hydrate/kg of `free' water ˆ kg of anhydrous substance/m3 of solution ˆ kg of hydrate/m3 of solution ˆ kmol of anhydrous substance/m3 of solution ˆ kmol of hydrate/m3 of solution ˆ mole fraction of anhydrous substance ˆ mole fraction of hydrate ˆ molar mass of anhydrous substance ˆ molar mass of hydrate ˆ molar mass of water ˆ MH /MA ˆ density of solution (kg m 3 )

C6 MA C7 R C6  MA C7



C6 C6 C6 R



MH C7 MH C7 MA C7 MH C7

C6 MH

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Interconversions between mass or molar units and those based on mole fractions are a little more complex than those in Table 3.1. The mole fraction x of a particular component in a mixture of several substances is given by x1 ˆ

m1 /M1 m1 /M1 ‡ m2 /M2 ‡ m3 /M3 ‡




(3:1)

where m is the mass of a particular component, and M its molar mass. The relationship between compositions expressed in mole fractions and in other units is given by C9 ˆ

MW C 1 MW C 4 ˆ MW C1 ‡ MA MH ‡ (MH ‡ MW

MA )C4

(3:2)

and C10 ˆ

MA

M W C1 (MH MA

MW )C1

ˆ

M W C4 M W C 4 ‡ MH

(3:3)

A large number of terms have been used to express the relative solubility of a solute in a given solvent. The following, together with some examples of solubility (g L 1 ) at around ambient temperature, are the most frequently encountered: Practically insoluble Slightly soluble Sparingly soluble Soluble Very soluble

BaSO4 Ca(OH)2 PbCl2 NaCl Sucrose

0.002 1.5 10 350 2000

3.4 Solubility correlations In the majority of cases the solubility of a solute in a solvent increases with temperature, but there are a few well-known exceptions to this rule. Some typical solubilities for various salts in water are shown in Figure 3.1, where all concentrations are expressed as kg of anhydrous substance per 100 kg of water. In Figure 3.1a sodium chloride is a good example of a salt whose solubility increases only slightly with an increase in temperature, whereas sodium acetate shows a fairly rapid increase. The solubility characteristics of a solute in a given solvent have a considerable influence on the choice of a method of crystallization. It would be useless, for instance, to cool a hot saturated solution of sodium chloride in the hope of depositing crystals in any quantity. A reasonable yield could only be achieved by removing some of the water by evaporation, and this is what is done in practice. On the other hand, a direct cooling crystallization operation would be adequate for a salt such as copper sulphate: cooling from 90 to 20  C would produce about 44 kg of CuSO4 for every 100 kg of water present in the original solution. As the stable phase of copper sulphate at 20  C is the pentahydrate the

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93

Figure 3.1. Solubility curves for some salts in water: (a) smooth curves, (b) indicating occurrence of phase changes

actual crystal yield would be about 69 kg of CuSO4
5H2 O for every 100 kg of water present initially. Not all solubility curves are smooth, as can be seen in Figure 3.1b. A discontinuity in the solubility curve denotes a phase change. For example, the solid phase deposited from an aqueous solution of sodium sulphate below 32.4  C will consist of the decahydrate, whereas the solid deposited above this temperature will consist of the anhydrous salt. The solubility of anhydrous sodium sulphate decreases with an increase in temperature. This negative solubility effect, or inverted solubility as it is sometimes called, is also exhibited by substances such as calcium sulphate (gypsum), calcium, barium and strontium acetates, calcium hydroxide, etc. These substances can cause trouble in certain types of crystallizer by causing a deposition of scale on heat-transfer surfaces. The solubility curves for two different phases meet at the transition point, and a system may show a number of these points. For instance, three forms of ferrous sulphate may be deposited from aqueous solution depending upon the temperature: FeSO4
7H2 O up to 56  C, FeSO4
4H2 O from 56 to 64  C and FeSO4
H2 O above 64  C. The general trend of a solubility curve can be predicted from Le Chatelier's Principle which, for the present purpose, can be stated: when a system in equilibrium is subjected to a change in temperature or pressure, the system will adjust itself to a new equilibrium state in order to relieve the effect of the change. Most solutes dissolve in their near-saturated solutions with an absorption of heat (endothermic heat of solution) and an increase in temperature results in an increase in the solubility. An inverted solubility effect occurs when the solute dissolves in its near-saturated solution with an evolution of heat (exothermic heat of solution).

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Strictly speaking, solubility is also a function of pressure, but the effect is generally negligible in the systems normally encountered in crystallization from solution. In the purification of melt systems, however, pressure manipulation can be utilized for separating organic isomers (section 7.3.2). Many equations have been proposed for the correlation and prediction of solubility data. Some are better than others, but none has been found to be of general applicability. In any case, an experimentally determined solubility is undoubtedly preferred to an estimated value, particularly in systems that may contain impurities. Nevertheless, there is frequently a need for a simple mathematical expression of solubility to assist the recording and correlation of data. One of the most commonly used expressions of the influence of temperature on solubility is the polynomial c ˆ A ‡ Bt ‡ Ct2 ‡




(3:4)

where t is the temperature, e.g. in  C, and c is the solution composition, expressed in any convenient units. A, B, C, etc. are constants that depend on the units used. There is rarely any need to resort to higher-order polynomials for this empirical relationship. In addition to equation 3.4, a number of semi-empirical equations have been proposed for solubility correlation purposes, some of which are based on thermodynamic relationships relating to phase equilibria. Examples of some of the expressions that have found favour, at one time or another, are log x ˆ A ‡ BT

(3:5)

log x ˆ A ‡ BT ‡ CT

2

(3:6)

log x ˆ A ‡ BT

1

log x ˆ A ‡ BT

1

‡ CT

log x ˆ A ‡ BT

1

‡ C log T

(3:7) 2

(3:8) (3:9)

In all these relationships the solution composition x is expressed as mole fraction of solute and the temperature T is expressed in kelvins (K). The constants A, B and C in equations 3.4 to 3.9, of course, are not related to one another. Broul, NyÂvlt and SoÈhnel (1981) came to the conclusion that, when tested against solubility data from 70 inorganic salts in water, the accuracy of the twoconstant equations was consistently lower than that of the three-constant equations. However, they found that there was little to choose between individual equations in these two groups, but they did select equation 3.9 as being the most reliable. There is still considerable merit, however, in favour of equation 3.7 because of its simple form and its usefulness in the graphical estimation of transition points. Conventional solubility plots, such as those shown in Figure 3.1, can prove unreliable for this purpose when only a few data points are available, especially when the points lie on one or more different curves. Solubilities plotted in accordance with equation 3.7 are shown in Figure 3.2. Mole fraction concentrations x are recorded on the logarithmic abscissa and values of 103 T 1

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95

Figure 3.2. Alternative method for the graphical representation of solubility data

(because T 1 in the range 273±373 K is a very small quantity) are recorded on the right-hand linear ordinate scale. Alternatively, log±reciprocal graph paper can be used on which the temperature in degrees Celcius can be plotted directly, as on the left-hand ordinate in Figure 3.2. In Figure 3.1 the solubility of CuSO4 over the temperature range 0±100  C is represented by a smooth curve, and the solubilities of Na2 SO4 and Na2 CrO4 are represented by smooth curves that intersect at transition points. Several advantages of the log x versus T 1 plot shown in Figure 3.2 immediately become apparent. For example, the data for the above three salts lie on a series of straight lines, which greatly assists interpolation and allows transition points to be identified with some precision. It is easier, for example, to produce the two straight lines for sodium sulphate in Figure 3.2, to meet at 32.4  C than it is to extend the two corresponding curves in Figure 3.1. The two straight lines for CuSO4 intersect at about 67  C, which indicates a phase transition at this temperature; this transition between two different crystalline forms of the pentahydrate is not detected in Figure 3.1. Incidentally, the transition CuSO4
5H2 O „ CuSO4
3H2 O occurs at 95.9  C. Only two of the transitions for the sodium chromate system are indicated in Figure 3.2. There are actually three transition points in this system: 10H2 O „ 6H2 O (19:6  C), 6H2 O „ 4H2 O (26:6  C) and 4H2 O „ anhydrous (64:8  C). The solubility data for sodium acetate are included in Figure 3.2 to illustrate the fact that straight lines do not always result from this method of plotting. Curved lines are often obtained for highly soluble substances, or in regions where the temperature coefficient of solubility is high, or in cases where several hydrates can exist over a narrow range of temperature. It is possible, of course, that the curved portion of the sodium acetate line in the region of about 40±58  C could be a series of straight lines representing hydrates other than the trihydrate, but there is no evidence to support this view.

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3.5 Theoretical crystal yield If the solubility data for a substance in a particular solvent are known, it is a simple matter to calculate the maximum yield of pure crystals that could be obtained by cooling or evaporating a given solution. The calculated yield will be a maximum, because the assumption has to be made that the final mother liquor in contact with the deposited crystals will be just saturated. Generally, some degree of supersaturation may be expected, but this cannot be estimated. The yield will refer only to the quantity of pure crystals deposited from the solution, but the actual yield of solid material may be slightly higher than that calculated, because crystal masses invariably retain some mother liquor even after filtration. When the crystals are dried they become coated with a layer of material that is frequently of a lower grade than that in the bulk of the crystals. Impure dry crystal masses produced commercially are very often the result of inadequate mother liquor removal. Washing on a filter helps to reduce the amount of mother liquor retained by a mass of crystals, but there is always the danger of reducing the final yield by dissolution during the washing operation. If the crystals are readily soluble in the working solvent, another liquid in which the substance is relatively insoluble may be used. Alternatively, a wash consisting of a cold, near-saturated solution of the pure substance in the working solvent may be employed. The efficiency of washing depends largely on the shape and size of the crystals (see section 8.6.1). The calculation of the yield for the case of crystallization by cooling is quite straightforward if the initial concentration and the solubility of the substance at the lower temperature are known. The calculation can be complicated slightly if some of the solvent is lost, deliberately or accidentally, during the cooling process, or if the substance itself removes some of the solvent, e.g. by taking up water of crystallization. All these possibilities are taken into account in the following equations, which may be used to calculate the maximum yields of pure crystals under a variety of conditions. Let C1 ˆ initial solution concentration (kg anhydrous salt/kg solvent) C2 ˆ final solution concentration (kg anhydrous salt/kg solvent) W ˆ initial mass of solvent (kg) V ˆ solvent lost by evaporation (kg per kg of original solvent) R ˆ ratio of molar masses of hydrate and anhydrous salt Y ˆ crystal yield (kg) Substance crystallizes unsolvated (e.g. anhydrous salt) Total loss of solvent: No loss of solvent:

Y ˆ WC1 Y ˆ W(C1

Partial loss of solvent: Y ˆ W[C1

(3:10) (3:11)

C2 ) C2 (1

V)]

(3:12)

Substance crystallizes as a solvate Total loss of free solvent:

Y ˆ WRC1

(3:13)

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No loss of solvent:

Yˆ

WR(C1 C2 ) 1 C2 (R 1)

(3:14)

Partial loss of solvent:

Yˆ

WR[C1 C2 (1 V)] 1 C2 (R 1)

(3:15)

Equation 3.15 can, of course, be used as the general equation for all cases. Example 1 Calculate the theoretical yield of pure crystals that could be obtained from a solution containing 100 kg of sodium sulphate (mol: wt: ˆ 142) in 500 kg of water by cooling to 10  C. The solubility of sodium sulphate at 10  C is 9 kg of anhydrous salt per 100 kg of water, and the deposited crystals will consist of the decahydrate (mol: wt: ˆ 322). Assume that 2 per cent of the water will be lost by evaporation during the cooling process. R ˆ 322/142 ˆ 2:27 C1 ˆ 0:2 kg Na2 SO4 per kg of water C2 ˆ 0:09 kg Na2 SO4 per kg of water W ˆ 500 kg of water V ˆ 0:02 kg per kg of water present initially Substituting these values in equation 3.15 gives 500  2:27[0:2 0:09(1 0:02)] 1 0:09(2:27 1) Yield ˆ 143 kg Na2 SO4
10H2 O Yˆ

To determine the crystal yield from a vacuum crystallizer (section 7.5.3) it is necessary to estimate the amount of solvent evaporated, V. This depends on the heat made available during the operation of the crystallizer, i.e. the sum of the sensible heat drop of the solution, which cools from the feed temperature to the equilibrium temperature in the vessel, and the heat of crystallization liberated. The heat balance, therefore, will be VWv ˆ c(t1

t2 )W(1 ‡ C1 ) ‡ c Y

(3:16)

where, in addition to the symbols defined for equation 3.15, v ˆ enthalpy of vaporization of solvent (kJ kg 1 ) c ˆ heat of crystallization of product (kJ kg 1 ) t1 ˆ initial temperature of solution ( C) t2 ˆ final temperature of solution ( C) c ˆ mean specific heat capacity of solution (kJ kg 1 K 1 ) Substituting for the value of Y from equation 3.15 and simplifying Vˆ

c R(C1

C2 ) ‡ c(t1 t2 )(1 ‡ C1 )[1 C2 (R v [1 C2 (R 1)] c RC2

1)]

(3:17)

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Example 2 Estimate the yield of sodium acetate crystals (CH3 COONa
3H2 O) from a continuous vacuum crystallizer operating with an internal pressure of 15 mbar when it is supplied with 2000 kg h 1 of a 40 per cent aqueous solution of sodium acetate (0.4 kg of anhydrous salt in 0.6 kg water) at 80  C. The boiling point elevation of the solution may be taken as 11.5  C. Heat of crystallization of CH3 COONa
3H2 O, c Specific heat capacity of the solution, c Latent heat of vaporization of water at 15 mbar, v Boiling point of water at 15 mbar Operating temperature ˆ 17:5  ‡ 11:5  C Solubility at 29  C, C2 Initial concentration, C1 ˆ 0:4/0:6 Initial mass of water in feed, W ˆ 0:6  2000 Ratio of molar masses, R ˆ 136/82

ˆ 144 kJ kg 1 ˆ 3:5 kJ kg 1 K ˆ 2:46 MJ kg 1 ˆ 17:5  C ˆ 29  C ˆ 0:539 kg kg 1 ˆ 0:667 kg kg 1 ˆ 1200 kg h 1 ˆ 1:66

1

The quantity of water vaporized is calculated from equation 3.17: V ˆ 0:153 kg/kg of water present originally which, when substituted in equation 3.15, gives the crystal yield as Y ˆ 660 kg h 1 of sodium acetate trihydrate.

3.6 Ideal and non-ideal solutions An ideal solution is one in which the interaction between solute and solvent molecules is identical with that between the solute molecules and the solvent molecules themselves. From this definition alone it is clear that a truly ideal solution is most unlikely to exist, but the concept is still very useful as a reference condition. For instance, if the solute and solvent did form an ideal solution, the solubility could be predicted from the van't Hoff equation:  
Hf 1 1 ln x ˆ (3:18) R Tf T where x is the mole fraction of the solute in the solution, T is the solution temperature (K), Tf is the fusion temperature (melting point) of the solute (K),
Hf is the molal enthalpy of fusion of the solute (J mol 1 ) and R is the gas constant (8:314 J mol 1 K 1 ). For example, the solubility of naphthalene at 20  C in an ideal solution may be calculated from its melting point (80  C) and enthalpy of fusion (18:8 kJ mol 1 ) to give x ˆ 0:269. In principle, therefore, by performing such calculations over a range of temperatures, an `ideal' solubility curve may be constructed, but it is important to note that any such calculated solubility is expressed without reference to any particular solvent. Furthermore, the assumption of ideality for most real solutions is generally unjustified. The potential unreliability of equation 3.18 in predicting solubility can be demonstrated by comparing the above calculated ideal solubility (x ˆ 0:269) of naphthalene with measured solubilities in a few common solvents:

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benzene 0:241

toluene 0:224

CC14 0:205

99

hexane 0:090

Even in the case of benzene, with its chemical similarity to naphthalene, ideality is barely approached (the predicted solubility is some 12% too high). The solution in hexane, however, is highly non-ideal (the prediction is 200% too high). The van't Hoff equation can also be written in the form: ln x ˆ


Hf
Sf ‡ RT R

(3:18a)

since
Hf ˆ Tf
Sf :
Sf is the molal entropy of fusion. It should be understood, however, that even though a plot of 1n x versus T 1 may give a straight line, its slope may differ from
Hf /R if the solution exhibits non-ideal behaviour. In such cases, the enthalpy and entropy of mixing must be taken into account by replacing
Hf with
Hd (of dissolution) and
Sf by
Sd (Beiny and Mullin, 1987), i.e., using: ln x ˆ


Hd
Sd ‡ RT R

(3:18b)

Another approach stems from a consideration of the Gibbs free energy change
G for a dissolution process, which in general may be expressed in terms of the enthalpy and entropy changes associated with the mixing process:
G ˆ
H

T
S

(3:19)

For the formation of an ideal solution, e.g. by mixing two liquids, the Gibbs free energy may also be expressed as
G ˆ RT ln x

(3:20)

where x is the solution composition expressed as a mole fraction of one of the components. The entropy change accompanying this dissolution process, from equation 3.19, is
S ˆ

R ln x

(3:21)

since the enthalpy of mixing,
H, is zero for an ideal solution. The overall free energy change may also be expressed in terms of the activity, a, of one of the components:
G ˆ RT ln a

(3:22)

In other words, if the solution is ideal, aˆx

(3:23)

For non-ideal solutions, however, equation 3.23 has to be modified by the appropriate activity coefficient, , i.e. a ˆ x

(3:24)

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Similarly, equation 3.18 should be expressed as  
Hf 1 1 ln (x ) ˆ R Tf T

(3:25)

The key to the use of equation 3.25 for the prediction of solubilities in non-ideal systems is a reliable estimation of the activity coefficient . For organic solutes in organic solvents, this may be achieved (Gmehling, Anderson and Prausnitz, 1978) by the UNIFAC group contribution method which is discussed, together with other techniques for the prediction of solubility data, in section 3.10. 3.6.1 Activity and ionic strength The colligative properties of solutions, e.g. osmotic pressure, boiling point elevations, freezing point depression and vapour pressure reduction, depend on the effect of solute concentration on the solvent activity. The chemical potential, , of a non-electrolyte in dilute solution may be expressed by  ˆ oc ‡ RT ln c

(3:26)

where c is the solute molar concentration (mol L 1 ) and oc is the standard chemical potential also expressed on a molar basis (J mol 1 ). Equation 3.26 is unsatisfactory for non-electrolyte solute concentrations in excess of about 0.1 molar, and it cannot be applied to solutions of electrolytes for concentrations greater than about 10 3 molar. For such cases an expression based on activity rather than concentration should be applied, e.g.  ˆ oc ‡ RT ln ac

(3:27)

where ac is the solute activity expressed on a molar basis, which is related to composition c through the corresponding activity coefficient, c , by ac ˆ c c

(3:28)

The activity coefficient becomes unity at infinite dilution, i.e. when ideality may be assumed. For electrolyte solutions it is more appropriate to use the mean ionic activity, a , defined with respect to the mean ionic concentration and mean ionic activity coefficient by ac ˆ ac ˆ (c c ) ˆ (Qc c )

(3:29)

where  is the number of moles of ions in 1 mole of electrolyte, i.e.  ˆ ‡ ‡ 

(3:30)

Q ˆ (‡‡   )1/

(3:31)

and

For non-electrolytes,  ˆ 1.

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101

From the Debye±HuÈckel theory of electrolytes, the limiting (infinite dilution) law gives the mean activity coefficient of the ion as log  ˆ

Ajz‡ z jI 1/2

where the ionic strength, I, expressed as mol L 1 , is defined by X I ˆ 12 ci z2i

(3:32)

(3:33)

where ci is the concentration (mol L 1 ) of the ith ionic species and zi the valency, and z‡ and z are the valencies of the cation and anion, respectively. The Debye±HuÈckel constant, A, has values of 0.493, 0.499, 0.509 and 0.519 at 5, 15, 25 and 35  C, respectively. Similar relationships to those of equations 3.26 to 3.33 may be written for solution compositions expressed as molality, m (mol/kg of solvent), and mole fraction, x, respectively. For example, equation 3.26 could be written as either  ˆ om ‡ RT ln m

(3:26a)

 ˆ ox ‡ RT ln x

(3:26b)

and equation 3.33 as either X Im ˆ 12 mi z2i

(3:33a)

or

or Ix ˆ 12

X

xi z2i

(3:33b)

For simplicity, however, only the molar-based ionic strength (I ˆ Ic ) will be used subsequently in this section. The Debye±HuÈckel limiting law (equation 3.32) has to be modified for all but the most dilute solutions, and many modifications have been proposed. For example, the GuÈntelberg equation:  1/2  I log  ˆ Ajz‡ z j (3:34) I ‡ I 1/2 is useful for solutions of sparingly soluble electrolytes, and the Davies equation:  1/2   I log  ˆ Ajz‡ z j 0:3I (3:35) I ‡ I 1/2 is generally quite satisfactory for values of I up to about 0:2 mol L 1 (Davies, 1962; Nancollas, 1966). For concentrated mixed electrolyte solutions, more complex relationships have to be employed (see section 3.10).

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The calculation of the ionic strength of a mixed solution of electrolytes may be illustrated, using equation 3.33, for a mixture of equal volumes of 0:1 mol L 1 NaCl and CaCl2 solutions, assuming complete dissociation: I ˆ 12([(Na‡ )  12 ] ‡ [(Ca2‡ )  22 ] ‡ [(Cl )  12 ]) ˆ 12 (0:1  1 ‡ 0:1  4 ‡ 3  0:1  1) ˆ 0:4 mol L

1

3.6.2 Association and dissociation For incompletely dissociated electrolytes, it is generally convenient to define another activity coefficient, the mean activity coefficient of ions in solution, 0 . For a binary electrolyte which dissociates according to M‡ A „ ‡ M z‡ ‡  Az

(3:36)

the concentration of free ions in a solution of molar concentration, c, is c, where  is the degree of dissociation, and the activity coefficients are related by 0 c c ˆ c c

(3:37)

0  ˆ c / c

(3:38)

i.e.

The degree of dissociation, , of dissolved electrolyte was first expressed by Arrhenius in 1887 as the ratio of the molar conductivity (see section 3.6.3) of the solution, , to that of a solution at its most extreme dilution, 0 , i.e.  ˆ /0

(3:39)

The degree of dissociation can also be expressed in terms of the van't Hoff factor, i, and the number of moles of ions, , in one mole of solute: iˆ1

 ‡ 

(3:40)

1 1

(3:41)

i.e. ˆ

i 

For strong electrolytes (virtually complete dissociation,  ! 1) iˆ

(3:42)

The dissociation of an electrolyte molecule in solution into oppositely charged ions, however, is by no means a simple matter. The ionic association theory, first developed by Bjerrum in 1926, indicates that some kind of association will still exist between oppositely charged ions even when they are several molecular diameters apart. The rates of dissociation and reformation, of molecules or other complexes, are extremely fast and it is doubtful if the ions can

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ever truly be considered to be `free'. Ion association should be taken into account in any complete treatment of aqueous electrolyte solutions since it reduces the value of the ionic activity coefficients. 3.6.3 Conductance and conductivity The current flowing through an electrolyte solution is proportional to the reciprocal of its resistance, R (Ohm's law). This quantity, 1/R, is called the conductance of the solution, which formerly was expressed as reciprocal ohms (mhos, 1 ), but now bears the SI unit name of the siemens, S. Thus, a solution with a resistance of 10 has a conductance of 0.1 S. The resistances of different solutions can be compared through the quantity known as the resistivity, , defined by the relationship  ˆ Ra/l

(3:43)

where l is the length of the conductivity path in the solution and a is its crosssectional area. The units of  are the ohm metre, m, or in SI units, siemens 1 metre, S 1 m. In a similar manner to resistances, the conductances can be compared through the quantity known as the conductivity,  ( 1 m 1 or S m 1 ), which is the reciprocal of the resistivity, i.e. ˆ

1 

(3:44)

For electrolyte solutions the molar conductivity,  (the conductivity per mole of electrolyte), is a useful characteristic since it allows comparisons to be made between solutions of different substances.  is defined by  ˆ /c

(3:45)

where c is the solution concentration, expressed as mol m 3 , giving  the units of 1 m2 mol 1 , i.e. S m2 mol 1 . According to Kohlrauch's law, the value of  at infinite dilution, 0 , is the sum of the corresponding molar ionic conductivities, also at infinite dilution (see Tables 3.2 and A.13), i.e. 0 ˆ  ‡ 0 ‡ 0

(3:46)

The utility of the above relationships in estimating the solubility of sparingly soluble salts is discussed in section 3.9.3. Furthermore, in addition to its straightforward application, equation 3.46 can be used to calculate 0 for weak electrolytes, e.g. organic acids, from 0 values for their strong electrolyte salts. For example, 0 for acetic acid can be calculated (Moore, 1972) from values of sodium acetate, HCl and sodium chloride. At 25  C: 0 (HAc) ˆ 0 (NaAc) ‡ 0 (HCl) ˆ (91 ‡ 425 ˆ 388  10

0 (NaCl)

128)  10 4

S m2 mol

1

4

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Table 3.2. Some molar ionic conductivities at infinite dilution at 25  C Cation

104 ‡ 0 (S m2 mol 1 )

H‡ K‡ NH‡ 4 Ag‡ Na‡ Li‡ 2‡ 1 2 Mg 2‡ 1 2 Ca 2‡ 1 2 Sr 2‡ 1 2 Ba 2‡ 1 2 Pb

349.8 73.5 73.4 61.9 50.1 38.7 53.1 59.5 59.5 63.6 65.0

The values of 0 expressed here as 10

1 cm2 mol 1 .

Anion OH Br I Cl F NO3 ClO4 CH3 CO2 2 1 2CO3 2 1 2SO4 2 1 2CrO4 4

S m2 mol

1

104 0 (S m2 mol 1 ) 198.5 78.4 76.8 76.3 55.0 71.4 68.0 40.9 74.0 78.8 83.0

are equivalent to the former common units of

3.6.4 Solubility products The solubility of a sparingly soluble electrolyte in water is often expressed in terms of the concentration solubility product, Kc . To take the simplest case, if one molecule of such an electrolyte dissociates in solution into x cations and y anions according to the equation Mx Ay „ xM z‡ ‡ yAz

(3:47)

where z‡ and z are the valencies of the ions, then for a saturated solution (c‡ )x (c )y ˆ constant ˆ Kc

(3:48)

where c‡ and c are the ionic concentrations expressed as mol L 1 . Solubility products are often recorded, for convenience, as pK values where pKc ˆ log Kc . Thus a value of Kc ˆ 3:9  10 6 would be reported as pKc ˆ 5:4. For a salt which produces two ions per molecule (1±1, 2±2, etc. electrolytes, i.e. for x ˆ y ˆ 1) c‡ ˆ c ˆ c , where c is the equilibrium solubility (mol L 1 ). Therefore, equation 3.48 becomes c ˆ (Kc )1/2

(3:49)

In general c ˆ (Kc /xx yy )1/(x‡y)

(3:50)

Therefore, for a 2±1 electrolyte c ˆ (Kc /4)1/3

(3:51)

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and for a 3±1 electrolyte c ˆ (Kc /27)1/4

(3:52)

For example, the solubility products of silver bromide, lead iodide and aluminium hydroxide at 18  C are 4:1  10 13 , 9:3  10 9 and 1:1  10 15 respectively, so their solubilities in water at this temperature may be expressed as: AgBr (equation 3.49): c ˆ (4:1  10

13 1/2

)

ˆ 6:4  10

7

mol L

1

PbI2 (equation 3.51): c ˆ [(7:5  10 9 )/4]1/3 ˆ 1:2  10

3

mol L

1

Al(OH)3 (equation 3.52): c ˆ [(1:1  10

15

)/27]1/4 ˆ 8:0  10

5

mol L

1

However, the simple solubility product principle has extremely limited use. It should, for example, be restricted to solutions of very sparingly soluble salts (< 10 3 mol L 1 ). For more concentrated solutions it is necessary to adopt a more fundamental approach involving the use of activity concepts. The activity solubility product, Ka , is defined (a‡ )x (a )y ˆ constant ˆ Ka

(3:53)

where a‡ and a are the ionic activities. As the activity of an ion may be expressed in terms of the ionic concentration, c, and the ionic activity coefficient, , equation 3.53 may be written (c‡ ‡ )x (c )y ˆ Ka

(3:54)

Ka ˆ Kc (  )

(3:55)

i.e.

where  is the mean ionic activity coefficient and  (ˆx ‡ y) is the number of moles of ions produced by one mole of electrolyte. So the concentration solubility product is equal to the activity solubility only when  ˆ 1, i.e. at infinite dilution. In practice, Ka and Kc may be assumed approximately equal for concentrations up to about 10 3 mol L 1 , but above this concentration significant deviations can occur. The activity of an ion depends on the concentration of all the other ions in solution, so the presence of a dissolved foreign electrolyte can greatly influence the value of  of a sparingly soluble salt. A number of cases which appear anomalous when the simple solubility product is used can be explained when activity coefficients are taken into account. For instance, the addition of a common ion generally decreases the solubility of a salt, but cases are known where additions of a salt with a common ion result in increase in solubility. The reason for this is that a large

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Figure 3.3. Relative increase in solubility with increase in ionic strength: (a) BaSO4 in KNO3 solution, (b) AgCl in KNO3 solution. (After Lewin, 1960)

increase in ionic concentration can cause a reduction in the activity coefficients. Thus from equation 3.54 an increase in c will result in a decrease in c‡ , i.e. precipitation of the sparingly soluble salt, if ‡ and remain fairly constant, but an increase in c to a value which reduces both ‡ and must result in an increase in c‡ if Ka is to remain constant. The addition of a salt without a common ion often increases the solubility; this again is the result of the increased ionic concentration reducing the activity coefficients (Figure 3.3). Lewin (1960) has pointed out that even equation 3.54 does not represent the true situation since a saturated aqueous solution in equilibrium with the solid phase involves the reversible reaction solid ‡ water „ saturated solution so at constant temperature and pressure Mx Ay ‡ (xb ‡ yc)H2 O „ x(M z‡
bH2 O) ‡ y(Az
cH2 O) solid

solvent

saturated solution

where b and c are the numbers of water molecules associated with the cation and anion respectively. Consequently, under equilibrium conditions, by the Law of Mass Action: (a‡ )x (a )y (asolid )(awater )(xb‡yc)

ˆK

(3:56)

which Lewin called the comprehensive activity solubility product. The use of this thermodynamically rigorous form of the solubility product can resolve most apparently anomalous problems.

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Common and diverse ion effects The addition of an electrolyte to a saturated solution of a sparingly soluble salt with a common ion depresses the solubility of the latter (the common ion effect) and leads to its precipitation. For example, the solubility product of AgCl at 25  C is 1:56  10 10 (Ka ˆ Kc at this dilution), i.e. [Ag‡ ][Cl ] ˆ 1:56  10

10

giving, from equation 3.49, a solubility of c ˆ 1:25  10

5

mol L

1

If a small quantity of a more soluble chloride, e.g. NaCl, is added to a saturated solution of AgCl the Cl concentration will temporarily exceed 1:25  10 5 mol L 1 , i.e. [Ag‡ ][Cl ] > 1:56  10

10

This unstable condition cannot persist so the system readjusts itself until the new ionic product equals the solubility product and this results in the precipitation of some of the AgCl. The solubility product principle can only be strictly applied to equilibrium conditions, although it has often been used to explain such precipitations as those encountered in qualitative analysis by the traditional wet-test methods. However, these sudden precipitations do not take place under anything like equilibrium conditions and the fact that reasonably successful predictions can usually be made is mainly due to the enormous excess ionic concentrations (supersaturations) generated compared with those required by the corresponding solubility products. Errors of magnitude of 105 107 per cent have been estimated (Lewin, 1960) for such calculations and these clearly swamp other variations such as neglect of solute activity coefficients, complex ion formation, etc. Whereas the presence in solution of an ion in common with a sparingly soluble salt can significantly decrease the salt solubility, the presence of an ion not in common with any of those of the solute can increase the solute solubility on account of the increase in ionic strength. For example, the solubility of silver bromide in water is increased by around 30% in a 0:1 mol L 1 sodium nitrate aqueous solution, as can be seen in the following rough calculation. The solubility of silver bromide in water at 15  C is about 6  10 7 mol L 1 . At this low concentration the `activity' and `concentration' solubility products may be assumed to be equal (see equation 3.55), i.e. Kc ˆ (c )2 ˆ 3:6  10

13

ˆ Ka

For a 0.1 molar solution of sodium nitrate, using equations 3.35 with a value of A ˆ 0:499 appropriate for 15  C, the calculated ionic activity coefficient ‡ is 0.783, from which it may be estimated that Kc ˆ 3:6  10

13

/(0:783)2 ˆ 5:87  10

13

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giving the solubility c ˆ Kc1/2 ˆ 7:7  10

7

mol L

1

Temperature effects The ionic product of water Kw ˆ [H‡ ][OH ]

(3:57)

is temperature dependent, rising from about 10 15 at 0  C, to 10 14 at 25  C and 10 13 at 60  C. Such a large variation is of considerable significance in the precipitation of sparingly soluble metal hydroxides, and its neglect can lead to gross errors. Fe(III) hydroxide, for example, dissociates according to: Fe(OH)3 „ Fe3‡ ‡ 3OH and the solubility product Kc ˆ [Fe3‡ ][OH ]3 i.e. [Fe3‡ ] ˆ

Kc Kc [H‡ ]3 ˆ [OH ]3 [Kw ]3

Thus at a given pH the Fe3‡ concentration above which precipitation may be considered possible is [Fe3‡ ] / Kw 3 A value of Kw ˆ 10 14 is commonly used in rough calculations, but this is only correct at 25  C. If it were to be used for conditions at 15  C, where the correct value of Kw is 0:45  10 14 , an error of about (1/0:45) 3  100, i.e. > 1000 per cent, would be incurred.

3.7 Particle size and solubility The relationship between particle size and solubility, originally derived for vapour pressures in liquid±vapour systems by Thomson (who became Lord Kelvin in 1892) in 1871, utilized later by Gibbs, and applied to solid±liquid systems by Ostwald (1900) and Freundlich (1926) may be expressed in the form   c(r) 2M (3:58) ln  ˆ c RTr where c(r) is the solubility of particles of size (radius) r, c is the normal equilibrium solubility of the substance, R is the gas constant, T is absolute temperature,  is the density of the solid, M is the molar mass of the solid in solution and is the interfacial tension of the solid in contact with the solution.

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The quantity  represents the number of moles of ions formed from one mole of electrolyte (equation 3.30). For a non-electrolyte,  ˆ 1. Confusingly, but for understandable reasons, equation 3.58 is referred to in the literature by a variety of names such as the Gibbs±Thomson, Gibbs±Kelvin and Ostwald±Freundlich relation. For consistency, however, the designation `Gibbs±Thomson' will be used throughout this work. As a result of the particle size solubility effect, solution compositions may exceed greatly the normal equilibrium saturation value if the excess solute particles dispersed in the solution are very small. For most solutes in water, however, the solubility increase only starts to become significant for particle sizes smaller than about 1 mm. For example, for barium sulphate at 25  C: T ˆ 298 K, M ˆ 233 kg kmol 1 ,  ˆ 2,  ˆ 4500 kg m 3 , ˆ 0:13 J m 2 , R ˆ 8:3  103 J kmol 1 K 1 . Thus for a 1 mm, crystal (r ˆ 5  10 7 m), c/c ˆ 1:005 (i.e. 0.5% increase). For 0:1 mm, c/c ˆ 1:06 (i.e. 6% increase) and for 0:01 mm, c/c ˆ 1:72 (i.e. 72% increase). For a very soluble organic compound such as sucrose (M ˆ 342 kg kmol 1 ,  ˆ 1,  ˆ 1590 kg m 3 , ˆ 10 2 J m 2 ) the effect is similar: 1 mm (0.4% increase), 0:1 mm (4%), 0:01 mm (40%). All such calculated values, however, should be treated with caution, not only because of the unreliability of values but also because the Gibbs±Thomson effect may cease to be influential at extremely small crystal sizes (see Figure 3.4). For practical application to crystals, equation 3.58 could be more usefully redrafted in terms of particle size expressed as a convenient length parameter, L, coupled with an approximate overall shape factor, F:   c(L) 2FM ln  ˆ (3:59) c 3RTL For spheres and cubes F ˆ 6 (L ˆ diameter or length of side). For other shapes F > 6, e.g. for an octahedron F ˆ 7:35 (section 2.14.3). Strictly speaking, equation 3.58 should be expressed in terms of solution activities rather than concentrations (EnuÈstuÈn and Turkevich, 1960). Furthermore, it involves a number of assumptions that may not always be valid. For example, the solid±liquid interfacial tension (section 5.6) is implicitly assumed to be independent of particle size, and no account is taken of any ionization or

Figure 3.4. The effect of particle size on solubility

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dissociation of the solute in solution. This latter effect, discussed in detail by Jones and Partington (1915), leads to the relationship   c(r) 2M (3:60) ln  ˆ c (1  ‡ )RTr where  is the degree of dissociation. For the case of complete dissociation ( ˆ 1) equation 3.60 reduces to equation 3.58. Another fault of equation 3.58 is that it postulates an exponential increase in solubility to infinity with a reduction in particle size to zero: c(r) ˆ c exp (2M /RTr)

(3:61)

To overcome this anomaly, Knapp (1922) considered that the total surface energy (interfacial tension) of very small solid particles should be regarded as the sum of their `normal' surface energy plus the surface electrical charge that such particles would carry. From these considerations he derived for the case of isolated charged spheres an equation of the form c(r) ˆ c exp(Ar

1

Br 4 )

(3:62)

where A ˆ 2 M/RT and B ˆ q2 M/8RT, q being the electrical charge on the particle of radius r and  its dielectric constant. Equation 3.62 gives a curve of the type shown in Figure 3.4. Subsequent work has tended to lend support to Knapp's postulation and measurements made by Harbury (1946) indicated maximum:equilibrium solubility ratios of 3, 6 and 13 for salts KNO3 , KClO3 , K2 Cr2 O7 and Na2 SO4
10H2 O in water.

3.8 Effect of impurities on solubility So-called pure solutions are rarely encountered outside the analytical laboratory, and even then the impurity levels are usually well within detectable limits. Industrial solutions, on the other hand, are almost invariably impure, by any definition of the term, and the impurities present can often have a considerable effect on the solubility characteristics of the main solute. If to a saturated binary solution of A (a solid solute) and B (a liquid solvent) a small amount of the third component C (also soluble in B) is added, one of four conditions can result. First, nothing may happen, although this is comparatively rare, in which case the system remains in its original saturated state. Second, component C may react or otherwise combine or react chemically with A by forming a complex or compound, thus altering the whole nature of the system. In the third case, the presence of component C may make the solution super-saturated with respect to solute A, which would then be precipitated. In the fourth case, the solution may become unsaturated with respect to A. The terms `salting-out' and `salting-in' are commonly used to describe these last two cases, particularly when electrolytes are involved.

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The salting-out effect of an electrolyte added to an aqueous solution of a non-electrolyte, can often be represented by the empirical equation log

c0 ˆ kC ‡ c

(3:63)

where c and c0 are the equilibrium saturation concentrations (mol L 1 of the non-electrolyte in pure water and in a salt solution of concentration C (mol L 1 ), respectively. The constants k, called the salting parameter, and refer to one particular electrolyte and its effect on one particular non-electrolyte at a given temperature. This type of relationship, which often applies with a reasonable accuracy for low non-electrolyte concentrations and electrolyte concentrations up to 4 or 5 mol L 1 , is commonly employed to characterize the precipitation of proteins from aqueous solution using inorganic electrolytes (Bell, Hoare and Dunnill, 1983). Occasionally the presence of an electrolyte increases the solubility of a nonelectrolyte (negative value of k) and this salting-in effect is exhibited by several salts, with large anions and cations, which themselves are very soluble in water. Sodium benzoate and sodium p-toluenesulphonate are good examples of these hydrotropic salts and the phenomenon of salting-in is sometime referred to as hydrotropism. Values of the salting parameter for three salts applied to benzoic acid are NaCl (0.17), KCl (0.14) and sodium benzoate ( 0:22). Long and McDevit (1952) have made a comprehensive review of salting-in and saltingout phenomena. Halstead (1970) reported that potassium sulphate once dissolved would not recrystallize from aqueous solutions contaminated with traces of chromium(III) or iron(III) and suggested that these impurities prevented the nucleation of K2 SO4 crystals. Trace impurities can sometimes have highly unexpected effects on equilibrium solubility measurements. For example, the solubility of potassium sulphate is significantly lowered when measured by dissolving K2 SO4 crystals in water containing ppm traces of Cr(III) (Kubota et al., 1988). A measured value obtained under these circumstances, however, is only an apparent or pseudo-solubility, the value of which is determined by two competing rate processes, viz., the adsorption of Cr(III) species on the K2 SO4 crystals and the dissolution of the crystals. Interestingly, a false solubility measurement is also obtained when, instead of approaching equilibrium from the undersaturated state, it is approached from the supersaturated condition, e.g., by cooling a solution to deposit excess solute. In this case, the recorded pseudo-solubility is higher than the true equilibrium value because the Cr(III) species in solution adsorb on the depositing K2 SO4 crystals, retard their growth and prevent complete desupersaturation of the solution. The actual species adsorbed, one of the many possible hydroxo-Cr(III) complexes, depends on the particular salt added as impurity and the pH of the solution (Kubota et al., 1994). The magnitude of the decrease in the measured pseudo-solubilities depends on pH, impurity concentration and the particular salt of the impurity added. Similar patterns of behaviour are seen when traces of Fe(III) salts are used as the added impurity (Kubota et al., 1999). An account of the effects of trace impurities on crystal growth and dissolution processes is given in section 6.2.8.

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When considerable quantities of a soluble impurity are present in, or deliberately added to, a binary solution, the system may be assessed better in terms of three components, expressing the data on a triangular ternary equilibrium diagram (see section 4.6).

3.9 Measurement of solubility Innumerable techniques, of almost infinite variety, have been proposed at one time or another for the measurement of the solubility of solids in liquids. No single method can be identified, however, as being generally applicable to all possible types of system. The choice of the most appropriate method for a given case has to be made in the light of the system properties, the availability of apparatus and analytical techniques, the skill and experience of the operators, the precision required, and so on. The accuracy required of a solubility measurement depends greatly on the use that is to be made of the information. Requirements vary enormously. In some cases, a simple assessment of whether a substance is highly, moderately or sparingly soluble in a given solvent, with some rough quantification, may be quite sufficient. In others, very high precisions may be demanded. For most work, however, a precision of < 1% should be aimed for, and usually this is not too difficult to attain. Extensive reviews of the literature on the subject of experimental solubility determination have been made by Vold and Vold (1949) and Zimmerman (1952). Purdon and Slater (1946) give an excellent account of the determination of solubility in aqueous salt systems. The monographs of Blasdale (1927) and Teeple (1929) give comprehensive accounts of the problems encountered in measuring equilibria in complex multicomponent aqueous salt systems. Temperature control Constant temperature control is essential during all the experimental procedures for solubility determination, not only during equilibration, but also during the sampling of saturated solution for analysis. The allowable limits of temperature variation depend on the system under investigation and the required precision of the solubility measurement. Much greater care has to be taken when the solubility changes appreciably with a change in temperature. In the determination of the solubility at 25  C of, say, sodium chloride in water (Figure 3.1a), a variation of 0:1  C in the experimental temperature would allow for a potential precision of less than 0.01 per cent in the solubility, but the same temperature variation would allow for more than 1 per cent in the case of sodium sulphate (Figure 3.1b). For most general purposes, a thermostat precision of better than 0:1  C is normally adequate. It is essential that any thermometers, thermocouples, thermistors, etc. used in the thermostat bath and equilibrium cell are accurately calibrated with reference to a standard thermometer. This point cannot be emphasized too strongly.

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Agitation of solutions Agitation is generally necessary to bring liquid and solid phases into intimate contact and facilitate equilibration. Agitation with a stirrer in an open vessel is not normally recommended, on account of the potential loss of solvent by evaporation, but sealed-agitated vessels are commonly used. Agitation in tightly stoppered vessels, that are rocked, rotated or shaken whilst immersed in a thermostat bath, is also quite a popular method, particularly when many samples have to be tested at the same time. Sampling Once equilibrium has been attained, i.e. when the originally over- and undersaturated solutions are of equal composition, the mixture is allowed to stand for an hour or more, at the relevant constant temperature, to enable any finely dispersed solid particles to settle. The withdrawal of a sample of clear supernatant liquid for analysis can be effected in a number of ways, depending on the characteristics of the system. For example, a suitably warmed pipette, with the tip protected by a piece of cotton wool, glass wool or similar substance, is often quite adequate. The pipette may be warmed to the appropriate temperature by leaving it standing in a stoppered tube immersed in the thermostat bath. Alternatively, a variety of sintered glass filters can be utilized (Figure 3.5) (NyÂvlt, 1977). In all cases, several portions of solution should first be withdrawn and discarded to satisfy any possible capacity of the filter to adsorb solute from the saturated solution (Vold and Vold, 1949). The sample of saturated solution may then be analysed by any convenient technique (section 3.9.2). It is important to note that a weighed quantity, not a measured volume, of solution should be taken for analysis. Weighing should be carried out to 0:0005 g if possible, depending on the required precision.

Figure 3.5. Some sintered glass filters for separating an equilibrium solid phase from a saturated solution. (After NyÂvlt, 1977)

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Achievement of equilibrium The achievement of equilibrium presents one of the major experimental difficulties in solubility determination. Prolonged agitated contact is required between excess solid solute and solution at a constant temperature, usually for several hours. In some cases, however, contact for days or even weeks may be necessary. Viscous solutions and systems at relatively low temperatures often require long contact times and so do substances of low solubility. A check should always be made, if possible, on the accuracy of a solubility determination at a given temperature, by approaching equilibrium from both the under-saturated and over-saturated states. In the first method a quantity of solid, in excess of the amount required to saturate the solvent at the given temperature, is added to the solvent and the two are agitated until apparent equilibrium is reached, i.e. when the solution attains a constant composition. In the second method the same quantities of solute and solvent are mixed, but the system is then heated for about 20 minutes above the required temperature, if the solubility increases with temperature, so that most but not all of the solid is dissolved. The solution is then cooled and agitated for a long period at the given temperature while the excess solid is deposited and an apparent equilibrium is reached. If the two solubility determinations agree, it can be reasonably assumed that the result represents the true equilibrium saturation concentration at the given temperature. If they do not, more time has to be allowed. This important point always has to be borne in mind when solubility measurements have to be made in solutions containing impurities (section 3.8). Unless solubility data for specific industrial substances are required, both the solute and solvent should be of the highest purity possible. The solute particles should be reasonably small to facilitate rapid dissolution, but not too small that the excess particles will not settle readily in the saturated solution. Settling is generally desirable to allow solid and liquid phase samples to be taken, after equilibration, for separate analysis. In practice, a close-sieved crystal fraction in the 100±300 mm size range is generally suitable for most purposes. 3.9.1 Solution and solid phase analysis Tremendous advances have been made in the past few decades in both the range and sensitivity of the analytical methods now available. For the purpose of solubility measurement, solution compositions can be measured by any convenient analytical technique, among which may be listed: liquid chromatography (HPLC), spectroscopy (UV, IR, NMR and mass), differential scanning calorimetry (DSC), differential thermal analysis (DTA), thermogravimetric analysis (TGA), refractometry, polarimetry, and most recently capillary electrophoresis (Altria, 2000). For the identification of crystalline polymorphs, IR spectroscopy and X-ray diffraction are the most commonly used techniques, while a combination of DSC±TGA or DTA±TGA are useful for analysing solvates. Descriptions of these techniques will be found in most handbooks of chemical and physical analysis (e.g. Findlay, 1973; Matthews, 1985). Comprehensive

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accounts of modern instrumental methods of analysis is given by Willard, Merritt, Dean and Settle (1988) and Ewing (1997). When the dissolved substance is stable to heat, the mass present in a known mass of solution may often be determined by gentle evaporation to dryness, followed by heating to constant mass in an air oven at around 100  C. Great care must be taken in this procedure to avoid solids loss by spattering. Another problem is that of solution creeping up the sides of the evaporating dish and over the edge. Gentle evaporation on a water bath may be used with aqueous solutions, covering the evaporating dish with a large funnel to prevent dust contamination. For substances that are not stable on heating to dryness, e.g. hydrates, their concentration in solution may be determined in some cases by chemical analysis and in others by measuring some concentration-dependent physical property. The latter is often convenient when a large number of determinations have to be made. Solution density and refractive index are probably the two properties most commonly measured for this purpose and the many well-established techniques available are fully described in standard textbooks of practical physical chemistry. In all cases, however, it is necessary to prepare a calibration chart. The first step, therefore, is to make up a series of solutions of known strength and measure the physical property in question for these solutions. This may be done for a range of temperatures, so that a series of calibration curves can be constructed. The composition of an unknown solution may then be determined by measuring the property in question at a given temperature, usually a few degrees above the saturation temperature to avoid the possibility of crystallization. Alternatively, a weighed quantity of the unknown solution is diluted with a known mass of solvent before measurement. It is always advisable, incidentally, to measure the density of a saturated solution at the same time that the equilibrium saturation concentration is being measured, for the simple practical reason that density is the mass±volume conversion factor, and this quantity is frequently required in process calculations. SoÈhnel and Novotny (1985) have published an extensive compilation of concentration±density data for aqueous solutions of inorganic salts. The final step that has to be taken in a solubility determination, in order to complete the information, is to determine the composition of the solid phase that was in equilibrium with the solution at the given temperature, remembering that the stable phase can change appreciably over quite a short range of temperature, especially in hydrated systems. For example, in the determination of the solubility of sodium carbonate in water over the temperature range 10±50  C, it would be found that the stable solid phase is Na2 CO3
10H2 O up to 32.0  C, Na2 CO3
7H2 O between 32.0 and 35.4  C and Na2 CO3
H2 O above 35.4  C. A sample of the equilibrium solid phase may be separated from its saturated solution by means of sintered glass filters like those depicted in Figure 3.5. It is necessary to ensure that the separation is made at the appropriate temperature, e.g. by carrying out the operation with the filter immersed in the thermostat bath itself. No matter how efficiently the filtration is made, however, the

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recovered solid particles will always be wet with adhering mother liquor and this must be accounted for in any subsequent analysis. First, it is essential to separate the wet solid sample as quickly as possible and transfer it immediately to a weighing bottle which can be closed to minimize loss of solvent. For simple binary systems, i.e. one solute and one solvent, correction for adhering mother liquor may be made in the following manner. The damp solid is weighed (mass m1 ) and then dried to constant mass, m2 , at an appropriate temperature, taking care to avoid decomposition, dehydration of hydrates, etc. The amount of solvent present in the mother liquor, s (e.g. grams of solvent per gram of solution) is determined from the saturated solution analysis described above. The mass of mother liquor, m3 , originally retained in the damp solid sample may then be calculated from m3 ˆ (m1

m2 )/s

(3:64)

The actual composition of the equilibrium solid phase may then be calculated from a simple mass balance, using the respective compositions of the damp solid and the equilibrium mother liquor. For multi-component systems the composition of the equilibrium solid phase may be determined indirectly by the so-called `wet residues' method first proposed by Schreinemakers (1893) in which the need for solid±liquid separation by filtration, etc. is avoided. The experimental procedures, together with those of the alternative `synthetic complex' method, are fully described in section 4.6.5. 3.9.2 Measurement techniques The so-called `synthesis' methods of solubility determination involve the preparation of a solvent±solute mixture of known composition, initially containing excess solute. The complete dissolution of the solid phase is then observed, either when the mixture is subjected to slow controlled heating (the `polythermal' methods) or at constant temperature when small quantities of fresh solvent are sequentially added over a period of time (the `isothermal' methods). The disappearance of the solid phase can be observed visually or monitored by recording some appropriate physical or physicochemical property of the system. Polythermal methods Solute and solvent are weighed into a small (50±100 mL) glass vessel in proportions corresponding approximately to the composition of a saturated solution in the middle of the proposed operating temperature range. The objective is to have some solid phase in excess at the lowest temperature used and all in solution at the highest. The closed vessel is fitted with a calibrated thermometer graduated in increments of 0:1  C and a suitable stirrer, and the contents are heated gently until all the crystals have been dissolved. The clear solution is first cooled until it is nucleated. Then, under controlled conditions, the temperature is increased

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slowly ( 0:2  C/ min) until the last crystal dissolves. At this point the equilibrium saturation temperature  has been achieved. Repeat runs will enable  for a given solution composition to be determined with a precision of 0:1  C for solutions with a moderate to high temperature coefficient of solubility (section 3.2). An apparatus in which the controlled heating and cooling sequences demanded in the above technique is depicted in Figure 5.9 and described in section 5.3. Determination of the instant at which all the crystals have finally dissolved in a solution is most commonly made by visual observation. In principle, however, the monitoring of any concentration-sensitive physical or physicochemical property (refractive index, conductivity, density, vapour pressure, particle size distribution, etc.) can offer alternative procedures. NyÂvlt (1977), for example, has described how refractive index measurements may be used for this purpose, where the sequence of events is as drawn in Figure 3.6a. As the solution containing suspended crystals is heated, the refractive index of the solution increases as the crystals dissolve. At point A the last crystal dissolves, at the equilibrium saturation temperature  . Further increases in temperature lead to a slow decrease in refractive index. The reverse curve traces the cooling sequence, with nucleation occurring at point B, when the refractive index suddenly falls to point C and subsequently follows the equilibrium saturation curve. The co